1.1.1
Solve the system by using elementary row operations on the equations. Follow the systematic elimination procedure.
x1+4x2
=
5
4x1+5x2
=
13
Find the solution to the system of equations.
(7,3)
(Simplify your answer. Type an ordered pair.)
1.1.1
Solve the system by using elementary row operations on the equations. Follow the systematic elimination procedure.
x1+5x2
=
9
4x1+5x2
=
9
Find the solution to the system of equations.
(6,3)
(Simplify your answer. Type an ordered pair.)
1.1.2
Solve the system by using elementary row operations on the equations. Follow the systematic elimination procedure.
5x1+10x2
=
20
4x1+7x2
=
18
Find the solution to the system of equations.
(8,2)
(Simplify your answer. Type an ordered pair.)
1.1.2
Solve the system by using elementary row operations on the equations. Follow the systematic elimination procedure.
6x1+12x2
=
18
4x1+7x2
=
15
Find the solution to the system of equations.
(9,3)
(Simplify your answer. Type an ordered pair.)
1.1.3
Find the point
x1,x2
that lies on the line
x1+2x2=5
and on the line
x1x2=1.
See the figure.
x1
x2
The point
x1,x2
that lies on the line
x1+2x2=5
and on the line
x1x2=1
is
(1,2).
(Simplify your answer. Type an ordered pair.)
1.1.3
Find the point
x1,x2
that lies on the line
x1+2x2=10
and on the line
x1x2=1.
See the figure.
x1
x2
The point
x1,x2
that lies on the line
x1+2x2=10
and on the line
x1x2=1
is
(4,3).
(Simplify your answer. Type an ordered pair.)
1.1.13
Solve the system.
x1
 
3x3
=
7
4x1
+
4x2
+
x3
=
23
2x2
+
3x3
=
1
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
The unique solution of the system is
4,2,1.
(Type integers or simplified fractions.)
The system has infinitely many solutions.
The system has no solution.
1.1.13
Solve the system.
x1
 
6x3
=
22
2x1
+
4x2
+
x3
=
21
2x2
+
3x3
=
1
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
The unique solution of the system is
4,4,3.
(Type integers or simplified fractions.)
The system has infinitely many solutions.
The system has no solution.
1.1.26
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
64h
325
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrix is the augmented matrix of a consistent linear system if
h=10.
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
The matrix is the augmented matrix of a consistent linear system if
h.
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
The matrix is the augmented matrix of a consistent linear system for every value of h.
The matrix is not the augmented matrix of a consistent linear system for any value of h.
1.1.26
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
64h
323
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrix is the augmented matrix of a consistent linear system if
h=6.
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
The matrix is the augmented matrix of a consistent linear system if
h.
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
The matrix is the augmented matrix of a consistent linear system for every value of h.
The matrix is not the augmented matrix of a consistent linear system for any value of h.
1.1.28
Determine whether the statement below is true or false. Justify the answer.
Elementary row operations on an augmented matrix never change the solution set of the associated linear system.
Choose the correct answer below.
The statement is true. Elementary row operations are always applied to an augmented matrix after the solution has been found.
The statement is false. Interchanging two rows never changes the solution set of the associated linear system. However, replacing one row by the sum of itself and a multiple of another row can change the solution set of that system.
The statement is true. Each elementary row operation replaces a system with an equivalent system.
The statement is false. Interchanging two rows never changes the solution set of the associated linear system. However, scaling a row by a nonzero constant can change the solution set of that system.
1.1.31
Determine whether the statement below is true or false. Justify the answer.
The solution set of a linear system involving variables
x1,
...,
xn
is a list of numbers
s1, ..., sn
that makes each equation in the system a true statement when the values
s1,
...,
sn
are substituted for
x1,
...,
xn,
respectively.
Choose the correct answer below.
The statement is true. It is the definition of the solution set of a linear system.
The statement is true. The solution set of a linear system will have the same number of elements as the list of the variables in the system.
The statement is false. The given description is of a single solution of such a system. The solution set of the system consists of all possible solutions.
The statement is false. The solution set of such a system consists of all possible orderings of the numbers
s1,
...,
sn.
1.1.31
Determine whether the statement below is true or false. Justify the answer.
The solution set of a linear system involving variables
x1,
...,
xn
is a list of numbers
s1, ..., sn
that makes each equation in the system a true statement when the values
s1,
...,
sn
are substituted for
x1,
...,
xn,
respectively.
Choose the correct answer below.
The statement is true. The solution set of a linear system will have the same number of elements as the list of the variables in the system.
The statement is false. The solution set of such a system consists of all possible orderings of the numbers
s1,
...,
sn.
The statement is false. The given description is of a single solution of such a system. The solution set of the system consists of all possible solutions.
The statement is true. It is the definition of the solution set of a linear system.
1.2.2
Determine which matrices are in reduced echelon form and which others are only in echelon form.
a.
1000
0100
0011
b.
1000
0500
0011
c.
1200
0000
0010
0001
Is matrix a in reduced echelon form, echelon form only, or neither?
echelon form only
reduced echelon form
neither
Is matrix b in reduced echelon form, echelon form only, or neither?
reduced echelon form
echelon form only
neither
Is matrix c in reduced echelon form, echelon form only, or neither?
echelon form only
reduced echelon form
neither
1.2.2
Determine which matrices are in reduced echelon form and which others are only in echelon form.
a.
01111
00111
00001
00000
b.
1000
0100
0011
c.
1500
0000
0010
0001
Is matrix a in reduced echelon form, echelon form only, or neither?
echelon form only
reduced echelon form
neither
Is matrix b in reduced echelon form, echelon form only, or neither?
echelon form only
reduced echelon form
neither
Is matrix c in reduced echelon form, echelon form only, or neither?
echelon form only
reduced echelon form
neither
1.2.2
Determine which matrices are in reduced echelon form and which others are only in echelon form.
a.
1000
0100
0011
b.
11011
06066
00033
00002
c.
1600
0000
0010
0001
Is matrix a in reduced echelon form, echelon form only, or neither?
echelon form only
reduced echelon form
neither
Is matrix b in reduced echelon form, echelon form only, or neither?
reduced echelon form
echelon form only
neither
Is matrix c in reduced echelon form, echelon form only, or neither?
reduced echelon form
echelon form only
neither
1.2.2
Determine which matrices are in reduced echelon form and which others are only in echelon form.
a.
11011
06066
00044
00002
b.
1011
0111
0000
c.
1600
0000
0010
0001
Is matrix a in reduced echelon form, echelon form only, or neither?
echelon form only
reduced echelon form
neither
Is matrix b in reduced echelon form, echelon form only, or neither?
echelon form only
reduced echelon form
neither
Is matrix c in reduced echelon form, echelon form only, or neither?
reduced echelon form
echelon form only
neither
1.2.2
Determine which matrices are in reduced echelon form and which others are only in echelon form.
a.
01111
00111
00001
00000
b.
1011
0111
0000
c.
0000
1400
0010
0001
Is matrix a in reduced echelon form, echelon form only, or neither?
reduced echelon form
echelon form only
neither
Is matrix b in reduced echelon form, echelon form only, or neither?
reduced echelon form
echelon form only
neither
Is matrix c in reduced echelon form, echelon form only, or neither?
reduced echelon form
echelon form only
neither
1.2.5
Describe the possible echelon forms of a nonzero
2×2
matrix.
Select all that apply. (Note that leading entries marked with a
may have any nonzero value and starred entries
(*)
may have any value including zero.)
***
**
0*
0**
0*0
000
1.2.5
Describe the possible echelon forms of a nonzero
2×2
matrix.
Select all that apply. (Note that leading entries marked with a
may have any nonzero value and starred entries
(*)
may have any value including zero.)
0*
***
0*0
000
0**
**
1.2.6
Describe the possible echelon forms of a nonzero
6×2
matrix.
Select all that apply. (Note that leading entries marked with a
may have any nonzero value and starred entries
(*)
may have any value including zero.)
00000000000
0*0*0*0*0*
0*000000000
0*00000000
1.2.6
Describe the possible echelon forms of a nonzero
6×2
matrix.
Select all that apply. (Note that leading entries marked with a
may have any nonzero value and starred entries
(*)
may have any value including zero.)
00000000000
0*00000000
0*0*0*0*0*
0*000000000
1.2.7
Find the general solution of the system whose augmented matrix is given below.
1348
2657
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
x1=43x2
x2 is free
x3=3
x1=
x2 is free
x3 is free
x1=
x2=
x3=
The system has no solution.
1.2.7
Find the general solution of the system whose augmented matrix is given below.
1433
2814
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
x1=
x2 is free
x3 is free
x1=
x2=
x3=
x1=34x2
x2 is free
x3=2
The system has no solution.
1.2.8
Find the general solution of the system whose augmented matrix is given below.
1409
27015
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
x1=3
x2=3
x3 is free
x1=
x2 is free
x3 is free
x1=
x2=
x3=
The system has no solution.
1.2.8
Find the general solution of the system whose augmented matrix is given below.
1104
32015
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
x1=7
x2=3
x3 is free
x1=
x2=
x3=
x1=
x2 is free
x3 is free
The system has no solution.
1.2.9
Find the general solution of the system whose augmented matrix is given below.
0146
141421
Select the correct choice below and, if necessary, fill in any answer boxes to complete your answer.
x1=
x2 is free
x3 is free
x1=
x2=
x3=
x1=3+2x3
x2=6+4x3
x3 is free
The system has no solution.
1.2.9
Find the general solution of the system whose augmented matrix is given below.
0123
1315
Select the correct choice below and, if necessary, fill in any answer boxes to complete your answer.
x1=4+5x3
x2=3+2x3
x3 is free
x1=
x2=
x3=
x1=
x2 is free
x3 is free
The system has no solution.
1.2.11
Find the general solution of the system whose augmented matrix is given below.
3270
128280
96210
Choose the correct answer below.
x1=23x273x3
x2 is free
x3 is free
x1=3x2
x2=2x3
x3 is free
x1=3
x2=2
x3=7
The system has no solutions.
1.2.11
Find the general solution of the system whose augmented matrix is given below.
5370
106140
159210
Choose the correct answer below.
x1=35x275x3
x2 is free
x3 is free
x1=5x2
x2=3x3
x3 is free
x1=5
x2=3
x3=7
The system has no solutions.
1.2.11
Find the general solution of the system whose augmented matrix is given below.
2390
46180
69270
Choose the correct answer below.
x1=32x292x3
x2 is free
x3 is free
x1=2x2
x2=3x3
x3 is free
x1=2
x2=3
x3=9
The system has no solutions.
1.2.11
Find the general solution of the system whose augmented matrix is given below.
3240
96120
6480
Choose the correct answer below.
x1=23x243x3
x2 is free
x3 is free
x1=3x2
x2=2x3
x3 is free
x1=3
x2=2
x3=4
The system has no solutions.
1.2.13
Find the general solution of the system whose augmented matrix is given below.
120108
010047
000145
000000
Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer.
x1=
x2 is free
x3=
x4 is free
x5 is free
x1=
x2=
x3 is free
x4=
x5=
x1=11+4x5
x2=7+4x5
x3 is free
x4=54x5
x5 is free
The system is inconsistent.
1.2.13
Find the general solution of the system whose augmented matrix is given below.
150104
010031
000132
000000
Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer.
x1=3+12x5
x2=1+3x5
x3 is free
x4=23x5
x5 is free
x1=
x2=
x3 is free
x4=
x5=
x1=
x2 is free
x3=
x4 is free
x5 is free
The system is inconsistent.
1.2.13
Find the general solution of the system whose augmented matrix is given below.
120109
010035
000189
000000
Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer.
x1=102x5
x2=5+3x5
x3 is free
x4=98x5
x5 is free
x1=
x2=
x3 is free
x4=
x5=
x1=
x2 is free
x3=
x4 is free
x5 is free
The system is inconsistent.
1.2.13
Find the general solution of the system whose augmented matrix is given below.
170103
010023
000172
000000
Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer.
x1=
x2 is free
x3=
x4 is free
x5 is free
x1=
x2=
x3 is free
x4=
x5=
x1=20+7x5
x2=3+2x5
x3 is free
x4=27x5
x5 is free
The system is inconsistent.
1.2.13
Find the general solution of the system whose augmented matrix is given below.
170102
010031
000145
000000
Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer.
x1=
x2=
x3 is free
x4=
x5=
x1=10+17x5
x2=1+3x5
x3 is free
x4=54x5
x5 is free
x1=
x2 is free
x3=
x4 is free
x5 is free
The system is inconsistent.
1.2.13
Find the general solution of the system whose augmented matrix is given below.
140106
010021
000167
000000
Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer.
x1=
x2=
x3 is free
x4=
x5=
x1=5+2x5
x2=1+2x5
x3 is free
x4=76x5
x5 is free
x1=
x2 is free
x3=
x4 is free
x5 is free
The system is inconsistent.
1.2.13
Find the general solution of the system whose augmented matrix is given below.
130106
010097
000185
000000
Select the correct choice below and, if necessary, fill in the answer boxes to complete your answer.
x1=
x2=
x3 is free
x4=
x5=
x1=20+19x5
x2=7+9x5
x3 is free
x4=58x5
x5 is free
x1=
x2 is free
x3=
x4 is free
x5 is free
The system is inconsistent.
1.2.14
Find the general solution of the system whose augmented matrix is given below.
105064
018103
000010
000000
Select the correct choice below and, if necessary, fill in any answer boxes to complete your answer.
x1=
x2=
x3=
x4=
x5 is free
x1=
x2=
x3 is free
x4=
x5=
x1=4+5x3
x2=3+x48x3
x3 is free
x4 is free
x5=0
The system is inconsistent.
1.2.14
Find the general solution of the system whose augmented matrix is given below.
105048
016107
000010
000000
Select the correct choice below and, if necessary, fill in any answer boxes to complete your answer.
x1=8+5x3
x2=7+x46x3
x3 is free
x4 is free
x5=0
x1=
x2=
x3=
x4=
x5 is free
x1=
x2=
x3 is free
x4=
x5=
The system is inconsistent.
1.2.14
Find the general solution of the system whose augmented matrix is given below.
105047
013102
000010
000000
Select the correct choice below and, if necessary, fill in any answer boxes to complete your answer.
x1=
x2=
x3 is free
x4=
x5=
x1=7+5x3
x2=2+x43x3
x3 is free
x4 is free
x5=0
x1=
x2=
x3=
x4=
x5 is free
The system is inconsistent.
1.2.14
Find the general solution of the system whose augmented matrix is given below.
106074
013105
000010
000000
Select the correct choice below and, if necessary, fill in any answer boxes to complete your answer.
x1=4+6x3
x2=5+x43x3
x3 is free
x4 is free
x5=0
x1=
x2=
x3=
x4=
x5 is free
x1=
x2=
x3 is free
x4=
x5=
The system is inconsistent.
1.2.22
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
1h2
42010
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrix is the augmented matrix of a consistent linear system if
h=.
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
The matrix is the augmented matrix of a consistent linear system if
h5.
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
The matrix is the augmented matrix of a consistent linear system for every value of h.
The matrix is not the augmented matrix of a consistent linear system for any value of h.
1.2.22
Determine the value(s) of h such that the matrix is the augmented matrix of a consistent linear system.
1h3
2103
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrix is the augmented matrix of a consistent linear system if
h5.
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
The matrix is the augmented matrix of a consistent linear system if
h=.
(Use a comma to separate answers as needed. Type an integer or a simplified fraction.)
The matrix is the augmented matrix of a consistent linear system for every value of h.
The matrix is not the augmented matrix of a consistent linear system for any value of h.
1.2.25
Determine whether the statement below is true or false. Justify the answer.
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
Choose the correct answer below.
The statement is false. For each matrix, there is only one sequence of row operations that row reduces it.
The statement is true. The echelon form of a matrix is always unique, but the reduced echelon form of a matrix might not be unique.
The statement is true. It is possible for there to be several different sequences of row operations that row reduce a matrix.
The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.
1.2.25
Determine whether the statement below is true or false. Justify the answer.
In some cases, a matrix may be row reduced to more than one matrix in reduced echelon form, using different sequences of row operations.
Choose the correct answer below.
The statement is false. Each matrix is row equivalent to one and only one reduced echelon matrix.
The statement is true. It is possible for there to be several different sequences of row operations that row reduce a matrix.
The statement is false. For each matrix, there is only one sequence of row operations that row reduces it.
The statement is true. The echelon form of a matrix is always unique, but the reduced echelon form of a matrix might not be unique.
1.2.26
Determine whether the statement below is true or false. Justify the answer.
The echelon form of a matrix is unique.
Choose the correct answer below.
The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique.
The statement is true. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations.
The statement is false. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed.
The statement is true. The echelon form of a matrix is always unique, but the reduced echelon form of a matrix might not be unique.
1.2.26
Determine whether the statement below is true or false. Justify the answer.
The echelon form of a matrix is unique.
Choose the correct answer below.
The statement is true. The echelon form of a matrix is always unique, but the reduced echelon form of a matrix might not be unique.
The statement is false. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed.
The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique.
The statement is true. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations.
1.2.26
Determine whether the statement below is true or false. Justify the answer.
The echelon form of a matrix is unique.
Choose the correct answer below.
The statement is true. The echelon form of a matrix is always unique, but the reduced echelon form of a matrix might not be unique.
The statement is true. Both the echelon form and the reduced echelon form of a matrix are unique. They are the same regardless of the chosen row operations.
The statement is false. The echelon form of a matrix is not unique, but the reduced echelon form is unique.
The statement is false. Neither the echelon form nor the reduced echelon form of a matrix are unique. They depend on the row operations performed.
1.2.29
Determine whether the statement below is true or false. Justify the answer.
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
Choose the correct answer below.
The statement is true. If a linear system has both basic and free variables, then each basic variable can be expressed in terms of the free variables.
The statement is true. It is the definition of a basic variable.
The statement is false. A variable that corresponds to a pivot column in the coefficient matrix is called a free variable, not a basic variable.
The statement is false. Not every linear system has basic variables.
1.2.29
Determine whether the statement below is true or false. Justify the answer.
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
Choose the correct answer below.
The statement is true. If a linear system has both basic and free variables, then each basic variable can be expressed in terms of the free variables.
The statement is false. Not every linear system has basic variables.
The statement is true. It is the definition of a basic variable.
The statement is false. A variable that corresponds to a pivot column in the coefficient matrix is called a free variable, not a basic variable.
1.2.29
Determine whether the statement below is true or false. Justify the answer.
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
Choose the correct answer below.
The statement is false. Not every linear system has basic variables.
The statement is true. It is the definition of a basic variable.
The statement is true. If a linear system has both basic and free variables, then each basic variable can be expressed in terms of the free variables.
The statement is false. A variable that corresponds to a pivot column in the coefficient matrix is called a free variable, not a basic variable.
1.2.29
Determine whether the statement below is true or false. Justify the answer.
A basic variable in a linear system is a variable that corresponds to a pivot column in the coefficient matrix.
Choose the correct answer below.
The statement is false. Not every linear system has basic variables.
The statement is true. It is the definition of a basic variable.
The statement is false. A variable that corresponds to a pivot column in the coefficient matrix is called a free variable, not a basic variable.
The statement is true. If a linear system has both basic and free variables, then each basic variable can be expressed in terms of the free variables.
1.2.31
Determine whether the statement below is true or false. Justify the answer.
Finding a parametric description of the solution set of a linear system is the same as solving the system.
Choose the correct answer below.
The statement is true. Solving a linear system is the same as finding the solution set of the system. The solution set of a linear system can always be expressed using a parametric description.
The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has no more than one solution.
The statement is true. Regardless of whether a linear system has free variables, the solution set of the system can be expressed using a parametric description.
The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution.
1.2.31
Determine whether the statement below is true or false. Justify the answer.
Finding a parametric description of the solution set of a linear system is the same as solving the system.
Choose the correct answer below.
The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has no more than one solution.
The statement is true. Solving a linear system is the same as finding the solution set of the system. The solution set of a linear system can always be expressed using a parametric description.
The statement is false. The solution set of a linear system can only be expressed using a parametric description if the system has at least one solution.
The statement is true. Regardless of whether a linear system has free variables, the solution set of the system can be expressed using a parametric description.
1.2.34
Determine whether the statement below is true or false. Justify the answer.
A general solution of a system is an explicit description of all solutions of the system.
Choose the correct answer below.
The statement is true. After applying the row reduction algorithm and generating a general solution of a system, the rightmost column displays all of the particular solutions of that system.
The statement is true. The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system, leading to a general solution of a system.
The statement is false. A general solution is the result of an inconsistent system, which has no particular solution.
The statement is false. Each different choice of a free variable produces the same solution of the system.
1.2.34
Determine whether the statement below is true or false. Justify the answer.
A general solution of a system is an explicit description of all solutions of the system.
Choose the correct answer below.
The statement is true. After applying the row reduction algorithm and generating a general solution of a system, the rightmost column displays all of the particular solutions of that system.
The statement is true. The row reduction algorithm leads directly to an explicit description of the solution set of a linear system when the algorithm is applied to the augmented matrix of the system, leading to a general solution of a system.
The statement is false. A general solution is the result of an inconsistent system, which has no particular solution.
The statement is false. Each different choice of a free variable produces the same solution of the system.
1.2.35
Suppose a
4×6
coefficient matrix for a system has
four
pivot columns. Is the system consistent? Why or why not?
Choose the correct answer below.
There is a pivot position in each row of the coefficient matrix. The augmented matrix will have
seven
columns and will not have a row of the form
0000001
,
so the system is consistent.
There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have
seven
columns, must have a row of the form
0000001
,
so the system is inconsistent.
There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have
seven
columns, could have a row of the form
0000001
,
so the system could be inconsistent.
There is a pivot position in each row of the coefficient matrix. The augmented matrix will have
five
columns and will not have a row of the form
00001
,
so the system is consistent.
1.2.35
Suppose a
5×8
coefficient matrix for a system has
five
pivot columns. Is the system consistent? Why or why not?
Choose the correct answer below.
There is a pivot position in each row of the coefficient matrix. The augmented matrix will have
six
columns and will not have a row of the form
000001
,
so the system is consistent.
There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have
nine
columns, could have a row of the form
000000001
,
so the system could be inconsistent.
There is a pivot position in each row of the coefficient matrix. The augmented matrix will have
nine
columns and will not have a row of the form
000000001
,
so the system is consistent.
There is at least one row of the coefficient matrix that does not have a pivot position. This means the augmented matrix, which will have
nine
columns, must have a row of the form
000000001
,
so the system is inconsistent.
1.2.36
Suppose a system of linear equations has a
3×5
augmented matrix whose fifth column is not a pivot column. Is the system consistent? Why or why not?
To determine if the linear system is consistent, use the portion of the Existence and Uniqueness Theorem, shown below.
A linear system is consistent if and only if the rightmost column of the augmented matrix
is not
a pivot column. That is, if and only if an echelon form of the augmented matrix has
no row
of the form
00b
with b nonzero.
In the augmented matrix described above, is the rightmost column a pivot column?
No
Yes
In the echelon form of the augmented matrix, is there a row of the form
0000b
with b nonzero?
No
Yes
Therefore, by the Existence and Uniqueness Theorem, the linear system is
consistent.
1.2.36
Suppose a system of linear equations has a
3×5
augmented matrix whose fifth column is not a pivot column. Is the system consistent? Why or why not?
To determine if the linear system is consistent, use the portion of the Existence and Uniqueness Theorem, shown below.
A linear system is consistent if and only if the rightmost column of the augmented matrix
is not
a pivot column. That is, if and only if an echelon form of the augmented matrix has
no row
of the form
00b
with b nonzero.
In the augmented matrix described above, is the rightmost column a pivot column?
Yes
No
In the echelon form of the augmented matrix, is there a row of the form
0000b
with b nonzero?
Yes
No
Therefore, by the Existence and Uniqueness Theorem, the linear system is
consistent.
1.2.37
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
Choose the correct answer below.
The system is consistent because the augmented matrix will contain a row of the form
00b
with b nonzero.
The system is consistent because the augmented matrix is row equivalent to one and only one reduced echelon matrix.
The system is consistent because the rightmost column of the augmented matrix is not a pivot column.
The system is consistent because all the columns in the augmented matrix will have a pivot position.
1.2.37
Suppose the coefficient matrix of a system of linear equations has a pivot position in every row. Explain why the system is consistent.
Choose the correct answer below.
The system is consistent because the rightmost column of the augmented matrix is not a pivot column.
The system is consistent because all the columns in the augmented matrix will have a pivot position.
The system is consistent because the augmented matrix will contain a row of the form
00b
with b nonzero.
The system is consistent because the augmented matrix is row equivalent to one and only one reduced echelon matrix.
1.2.45
Suppose experimental data are represented by a set of points in the plane. An interpolating polynomial for the data is a polynomial whose graph passes through every point. In scientific work, such a polynomial can be used, for example, to estimate values between the known data points. Another use is to create curves for graphical images on a computer screen. One method for finding an interpolating polynomial is to solve a system of linear equations. Find the interpolating polynomial
p(t)=a0+a1t+a2t2
for the data
(1,12),
(2,15),
(3,16).
That is, find
a0,
a1,
and
a2
such that the following is true.
a0
+
a1(1)
+
a2(1)2
=
12
a0
+
a1(2)
+
a2(2)2
=
15
a0
+
a1(3)
+
a2(3)2
=
16
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The interpolating polynomial is
p(t)=7+6tt2.
There are infinitely many possible interpolating polynomials.
There does not exist an interpolating polynomial for the given data.
1.2.45
Suppose experimental data are represented by a set of points in the plane. An interpolating polynomial for the data is a polynomial whose graph passes through every point. In scientific work, such a polynomial can be used, for example, to estimate values between the known data points. Another use is to create curves for graphical images on a computer screen. One method for finding an interpolating polynomial is to solve a system of linear equations. Find the interpolating polynomial
p(t)=a0+a1t+a2t2
for the data
(1,11),
(2,19),
(3,21).
That is, find
a0,
a1,
and
a2
such that the following is true.
a0
+
a1(1)
+
a2(1)2
=
11
a0
+
a1(2)
+
a2(2)2
=
19
a0
+
a1(3)
+
a2(3)2
=
21
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The interpolating polynomial is
p(t)=3+17t3t2.
There are infinitely many possible interpolating polynomials.
There does not exist an interpolating polynomial for the given data.
1.2.45
Suppose experimental data are represented by a set of points in the plane. An interpolating polynomial for the data is a polynomial whose graph passes through every point. In scientific work, such a polynomial can be used, for example, to estimate values between the known data points. Another use is to create curves for graphical images on a computer screen. One method for finding an interpolating polynomial is to solve a system of linear equations. Find the interpolating polynomial
p(t)=a0+a1t+a2t2
for the data
(1,14),
(2,18),
(3,20).
That is, find
a0,
a1,
and
a2
such that the following is true.
a0
+
a1(1)
+
a2(1)2
=
14
a0
+
a1(2)
+
a2(2)2
=
18
a0
+
a1(3)
+
a2(3)2
=
20
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The interpolating polynomial is
p(t)=8+7tt2.
There are infinitely many possible interpolating polynomials.
There does not exist an interpolating polynomial for the given data.
1.3.5
Write a system of equations that is equivalent to the given vector equation.
x1
2
4
6
+x2
6
0
8
=
4
5
7
Choose the correct answer below.
2x1
+
6x2
=
4
4x1
+
x2
=
5
6x1
8x2
=
7
2x1
+
6x2
=
7
4x1
=
5
6x1
8x2
=
4
2x1
+
6x2
=
4
4x1
=
5
6x1
8x2
=
7
2x1
+
6x2
=
5
4x1
=
4
6x1
8x2
=
7
1.3.5
Write a system of equations that is equivalent to the given vector equation.
x1
4
4
7
+x2
6
0
8
=
2
2
9
Choose the correct answer below.
4x1
+
6x2
=
2
4x1
+
x2
=
2
7x1
8x2
=
9
4x1
+
6x2
=
2
4x1
=
2
7x1
8x2
=
9
4x1
+
6x2
=
2
4x1
=
2
7x1
8x2
=
9
4x1
+
6x2
=
9
4x1
=
2
7x1
8x2
=
2
1.3.6
Write a system of equations that is equivalent to the given vector equation.
x1
5
2
+x2
6
4
+x3
5
1
=
0
0
Choose the correct answer below.
5x1
+
6x2
+
5x3
=
0
2x1
+
4x2
+
x3
=
0
5x1
+
6x2
5x3
=
0
2x1
+
4x2
=
0
5x1
+
6x2
+
5x3
=
0
2x1
+
4x2
=
0
5x1
+
6x2
5x3
=
0
2x1
+
4x2
+
x3
=
0
1.3.6
Write a system of equations that is equivalent to the given vector equation.
x1
4
3
+x2
7
4
+x3
2
1
=
0
0
Choose the correct answer below.
4x1
+
7x2
2x3
=
0
3x1
+
4x2
+
x3
=
0
4x1
+
7x2
+
2x3
=
0
3x1
+
4x2
+
x3
=
0
4x1
+
7x2
+
2x3
=
0
3x1
+
4x2
=
0
4x1
+
7x2
2x3
=
0
3x1
+
4x2
=
0
1.3.9
Write a vector equation that is equivalent to the given system of equations.
x2
+
3x3
=
0
5x1
+
9x2
x3
=
0
x1
+
4x2
6x3
=
0
Choose the correct answer below.
x1
0
5
1
+x2
1
9
4
+x3
0
0
0
=
3
1
6
x1
1
9
4
+x2
0
5
1
+x3
3
1
6
=
0
0
0
x1
0
5
1
+x2
3
1
6
+x3
1
9
4
=
0
0
0
x1
0
5
1
+x2
1
9
4
+x3
3
1
6
=
0
0
0
1.3.9
Write a vector equation that is equivalent to the given system of equations.
x2
+
2x3
=
0
4x1
+
6x2
x3
=
0
x1
+
3x2
8x3
=
0
Choose the correct answer below.
x1
0
4
1
+x2
2
1
8
+x3
1
6
3
=
0
0
0
x1
0
4
1
+x2
1
6
3
+x3
2
1
8
=
0
0
0
x1
0
4
1
+x2
1
6
3
+x3
0
0
0
=
2
1
8
x1
1
6
3
+x2
0
4
1
+x3
2
1
8
=
0
0
0
1.3.9
Write a vector equation that is equivalent to the given system of equations.
x2
+
2x3
=
0
5x1
+
8x2
x3
=
0
x1
+
3x2
8x3
=
0
Choose the correct answer below.
x1
1
8
3
+x2
0
5
1
+x3
2
1
8
=
0
0
0
x1
0
5
1
+x2
1
8
3
+x3
0
0
0
=
2
1
8
x1
0
5
1
+x2
2
1
8
+x3
1
8
3
=
0
0
0
x1
0
5
1
+x2
1
8
3
+x3
2
1
8
=
0
0
0
1.3.9
Write a vector equation that is equivalent to the given system of equations.
x2
+
5x3
=
0
5x1
+
6x2
x3
=
0
x1
+
3x2
8x3
=
0
Choose the correct answer below.
x1
0
5
1
+x2
5
1
8
+x3
1
6
3
=
0
0
0
x1
0
5
1
+x2
1
6
3
+x3
5
1
8
=
0
0
0
x1
0
5
1
+x2
1
6
3
+x3
0
0
0
=
5
1
8
x1
1
6
3
+x2
0
5
1
+x3
5
1
8
=
0
0
0
1.3.11
Determine if b is a linear combination of
a1,
a2,
and
a3.
a1=
1
3
0
,
a2=
0
1
4
,
a3=
5
6
36
,
b=
2
1
20
Choose the correct answer below.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column.
Vector b is not a linear combination of
a1,
a2,
and
a3.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.
1.3.11
Determine if b is a linear combination of
a1,
a2,
and
a3.
a1=
1
4
0
,
a2=
0
1
3
,
a3=
5
6
42
,
b=
3
2
30
Choose the correct answer below.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column.
Vector b is not a linear combination of
a1,
a2,
and
a3.
1.3.11
Determine if b is a linear combination of
a1,
a2,
and
a3.
a1=
1
3
0
,
a2=
0
1
2
,
a3=
5
5
20
,
b=
4
1
22
Choose the correct answer below.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column.
Vector b is not a linear combination of
a1,
a2,
and
a3.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.
1.3.11
Determine if b is a linear combination of
a1,
a2,
and
a3.
a1=
1
2
0
,
a2=
0
1
4
,
a3=
6
5
28
,
b=
3
2
16
Choose the correct answer below.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column and the second entry in the second column.
Vector b is not a linear combination of
a1,
a2,
and
a3.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column.
Vector b is a linear combination of
a1,
a2,
and
a3.
The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column.
1.3.13
Determine if b is a linear combination of the vectors formed from the columns of the matrix A.
A=
134
025
3912
,
b=
4
5
4
Choose the correct answer below.
Vector b is a linear combination of the vectors formed from the columns of the matrix A. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the third column.
Vector b is a linear combination of the vectors formed from the columns of the matrix A. The pivots in the corresponding echelon matrix are in the first entry in the first column, the second entry in the second column, and the third entry in the fourth column.
Vector b is not a linear combination of the vectors formed from the columns of the matrix A.
Vector b is a linear combination of the vectors formed from the columns of the matrix A. The pivots in the corresponding echelon matrix are in the first entry in the first column and the third entry in the second column, and the third entry in the third column.
1.3.15
List five vectors in
Spanv1, v2.
Do not make a sketch.
v1=
6
1
7
,
v2=
5
3
0
List five vectors in
Spanv1, v2.
0
0
0
,
6
1
7
,
5
3
0
,
1
4
7
,
11
2
7
(Use the matrix template in the math palette. Use a comma to separate vectors as needed. Type an integer or a simplified fraction for each vector element. Type each answer only once.)
1.3.15
List five vectors in
Spanv1, v2.
Do not make a sketch.
v1=
6
2
7
,
v2=
4
4
0
List five vectors in
Spanv1, v2.
0
0
0
,
6
2
7
,
4
4
0
,
2
6
7
,
10
2
7
(Use the matrix template in the math palette. Use a comma to separate vectors as needed. Type an integer or a simplified fraction for each vector element. Type each answer only once.)
1.3.27
Determine whether the statement below is true or false. Justify the answer.
An example of a linear combination of vectors
v1
and
v2
is the vector
12v1.
Choose the correct answer below.
The statement is true because
12v1=14v1+14v2.
The statement is false because
12v1
cannot be expressed as a multiple of
v2.
The statement is true because
12v1=12v1+0v2.
The statement is false because a linear combination of
v1
and
v2
needs to involve both of those vectors.
1.3.32
Determine whether the statement below is true or false. Justify the answer.
Asking whether the linear system corresponding to an augmented matrix
a1a2a3b
has a solution amounts to asking whether
b
is in
Span a1, a2, a3.
Choose the correct answer below.
The statement is false. If
b
corresponds to the origin, then it cannot be in
Span a1, a2, a3.
The statement is true. The solution of the linear system corresponding to
a1a2a3b
is always in
Span a1, a2, a3.
The statement is true. The linear system corresponding to
a1a2a3b
has a solution when
b
can be written as a linear combination of
a1,
a2,
and
a3.
This is equivalent to saying that
b
is in
Span a1, a2, a3.
The statement is false. It is possible for the linear system corresponding to
a1a2a3b
to have a solution without
b
being in
Span a1, a2, a3.
1.3.32
Determine whether the statement below is true or false. Justify the answer.
Asking whether the linear system corresponding to an augmented matrix
a1a2a3b
has a solution amounts to asking whether
b
is in
Span a1, a2, a3.
Choose the correct answer below.
The statement is true. The linear system corresponding to
a1a2a3b
has a solution when
b
can be written as a linear combination of
a1,
a2,
and
a3.
This is equivalent to saying that
b
is in
Span a1, a2, a3.
The statement is false. If
b
corresponds to the origin, then it cannot be in
Span a1, a2, a3.
The statement is true. The solution of the linear system corresponding to
a1a2a3b
is always in
Span a1, a2, a3.
The statement is false. It is possible for the linear system corresponding to
a1a2a3b
to have a solution without
b
being in
Span a1, a2, a3.
1.3.32
Determine whether the statement below is true or false. Justify the answer.
Asking whether the linear system corresponding to an augmented matrix
a1a2a3b
has a solution amounts to asking whether
b
is in
Span a1, a2, a3.
Choose the correct answer below.
The statement is false. If
b
corresponds to the origin, then it cannot be in
Span a1, a2, a3.
The statement is true. The solution of the linear system corresponding to
a1a2a3b
is always in
Span a1, a2, a3.
The statement is true. The linear system corresponding to
a1a2a3b
has a solution when
b
can be written as a linear combination of
a1,
a2,
and
a3.
This is equivalent to saying that
b
is in
Span a1, a2, a3.
The statement is false. It is possible for the linear system corresponding to
a1a2a3b
to have a solution without
b
being in
Span a1, a2, a3.
1.3.32
Determine whether the statement below is true or false. Justify the answer.
Asking whether the linear system corresponding to an augmented matrix
a1a2a3b
has a solution amounts to asking whether
b
is in
Span a1, a2, a3.
Choose the correct answer below.
The statement is false. It is possible for the linear system corresponding to
a1a2a3b
to have a solution without
b
being in
Span a1, a2, a3.
The statement is true. The linear system corresponding to
a1a2a3b
has a solution when
b
can be written as a linear combination of
a1,
a2,
and
a3.
This is equivalent to saying that
b
is in
Span a1, a2, a3.
The statement is true. The solution of the linear system corresponding to
a1a2a3b
is always in
Span a1, a2, a3.
The statement is false. If
b
corresponds to the origin, then it cannot be in
Span a1, a2, a3.
1.3.33
Let
A=
106
035
695
and
b=
4
4
20
.
Denote the columns of A by
a1,
a2,
a3,
and let
W=Span a1, a2, a3.
a. Is
b
in
a1, a2, a3?
How many vectors are in
a1, a2, a3?
b. Is
b
in W? How many vectors are in W?
c. Show that
a1
is in W.
[Hint:
Row operations are unnecessary.]
a. Is
b
in
a1, a2, a3?
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
Yes,
b
is in
a1, a2, a3
since, although
b
is not equal to
a1,
a2,
or
a3,
it can be expressed as a linear combination of them. In particular,
b=a1+a2+a3.
(Simplify your answers.)
Yes,
b
is in
a1, a2, a3
since
b=a.
(Type a whole number.)
No,
b
is not in
a1, a2, a3
since it cannot be generated by a linear combination of
a1,
a2,
and
a3.
No,
b
is not in
a1, a2, a3
since
b
is not equal to
a1,
a2,
or
a3.
How many vectors are in
a1, a2, a3?
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
There is(are)
3
vector(s) in
a1, a2, a3.
(Type a whole number.)
There are infinitely many vectors in
a1, a2, a3.
b. Is
b
in W? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
No,
b
is not in W since
b
is not equal to
a1,
a2,
or
a3.
Yes,
b
is in W since
b=a.
(Type a whole number.)
Yes,
b
is in W since, although
b
is not equal to
a1,
a2,
or
a3,
it can be expressed as a linear combination of them. In particular,
b=2a1+3a2+1a3.
(Simplify your answers.)
No,
b
is not in W since it cannot be generated by a linear combination of
a1,
a2,
and
a3.
How many vectors are in W? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
There is(are)
vector(s) in W.
(Type a whole number.)
There are infinitely many vectors in W.
c. The vector
a1
is in W if and only if
a1
can be written in the form
c1a1+c2a2+c3a3
with
c1, c2, and c3 all being scalars.
The vector
a1
can be written as the linear combination
1a1+0a2+0a3.
This satisfies the condition identified in the previous step, so it follows that
a1
is in W.
(Simplify your answers.)
1.3.33
Let
A=
105
023
445
and
b=
7
4
2
.
Denote the columns of A by
a1,
a2,
a3,
and let
W=Span a1, a2, a3.
a. Is
b
in
a1, a2, a3?
How many vectors are in
a1, a2, a3?
b. Is
b
in W? How many vectors are in W?
c. Show that
a2
is in W.
[Hint:
Row operations are unnecessary.]
a. Is
b
in
a1, a2, a3?
Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
No,
b
is not in
a1, a2, a3
since it cannot be generated by a linear combination of
a1,
a2,
and
a3.
Yes,
b
is in
a1, a2, a3
since
b=a.
(Type a whole number.)
Yes,
b
is in
a1, a2, a3
since, although
b
is not equal to
a1,
a2,
or
a3,
it can be expressed as a linear combination of them. In particular,
b=a1+a2+a3.
(Simplify your answers.)
No,
b
is not in
a1, a2, a3
since
b
is not equal to
a1,
a2,
or
a3.
How many vectors are in
a1, a2, a3?
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
There is(are)
3
vector(s) in
a1, a2, a3.
(Type a whole number.)
There are infinitely many vectors in
a1, a2, a3.
b. Is
b
in W? Select the correct choice below and, if necessary, fill in the answer box(es) to complete your choice.
No,
b
is not in W since it cannot be generated by a linear combination of
a1,
a2,
and
a3.
No,
b
is not in W since
b
is not equal to
a1,
a2,
or
a3.
Yes,
b
is in W since
b=a.
(Type a whole number.)
Yes,
b
is in W since, although
b
is not equal to
a1,
a2,
or
a3,
it can be expressed as a linear combination of them. In particular,
b=3a1+1a2+2a3.
(Simplify your answers.)
How many vectors are in W? Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
There is(are)
vector(s) in W.
(Type a whole number.)
There are infinitely many vectors in W.
c. The vector
a2
is in W if and only if
a2
can be written in the form
c1a1+c2a2+c3a3
with
c1, c2, and c3 all being scalars.
The vector
a2
can be written as the linear combination
0a1+1a2+0a3.
This satisfies the condition identified in the previous step, so it follows that
a2
is in W.
(Simplify your answers.)
1.4.7
Use the definition of
Ax
to write the vector equation as a matrix equation.
x1
9
8
4
6
+x2
1
8
9
4
+x3
6
8
7
4
=
5
3
1
7
The matrix equation is
916
888
497
644
x1
x2
x3
=
5
3
1
7
1.4.7
Use the definition of
Ax
to write the vector equation as a matrix equation.
x1
2
5
6
2
+x2
0
4
4
7
+x3
2
7
2
2
=
9
8
1
3
The matrix equation is
202
547
642
272
x1
x2
x3
=
9
8
1
3
1.4.8
Use the definition of
Ax
to write the vector equation as a matrix equation.
z1
1
1
+z2
5
5
+z3
4
0
+z4
5
5
=
3
1
The vector equation written as a matrix equation is
1545
1505
z1
z2
z3
z4
=
3
1
.
1.4.8
Use the definition of
Ax
to write the vector equation as a matrix equation.
z1
4
3
+z2
2
1
+z3
3
5
+z4
2
5
=
3
18
The vector equation written as a matrix equation is
4232
3155
z1
z2
z3
z4
=
3
18
.
1.4.9
Write the system first as a vector equation and then as a matrix equation.
8x1
+
x2
3x3
=
8
6x2
+
10x3
=
0
Write the system as a vector equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.
x1
x2
x3
=
x1
8
0
+x2
1
6
+x3
3
10
=
8
0
x1x2x3
=
Write the system as a matrix equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.
x1x2x3
=
813
0610
x1
x2
x3
=
8
0
x1+x2+x3=
1.4.10
Write the system first as a vector equation and then as a matrix equation.
9x1
x2
=
5
4x1
+
5x2
=
4
10x1
x2
=
1
Write the system as a vector equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.
x1
x2
=
x1x2
=
x1
9
4
10
+x2
1
5
1
=
5
4
1
Write the system as a matrix equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.
x1x2
=
91
45
101
x1
x2
=
5
4
1
x1+x2=
1.4.10
Write the system first as a vector equation and then as a matrix equation.
2x1
x2
=
10
5x1
+
6x2
=
4
9x1
x2
=
1
Write the system as a vector equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.
x1
x2
=
x1
2
5
9
+x2
1
6
1
=
10
4
1
x1x2
=
Write the system as a matrix equation where the first equation of the system corresponds to the first row. Select the correct choice below and fill in any answer boxes to complete your choice.
21
56
91
x1
x2
=
10
4
1
x1+x2=
x1x2
=
1.4.11
Given A and b to the right, write the augmented matrix for the linear system that corresponds to the matrix equation
Ax=b.
Then solve the system and write the solution as a vector.
A=
123
252
324
,
b=
4
13
8
Write the augmented matrix for the linear system that corresponds to the matrix equation
Ax=b.
1234
25213
3248
Solve the system and write the solution as a vector.
x=
x1
x2
x3
=
6
5
0
1.4.11
Given A and b to the right, write the augmented matrix for the linear system that corresponds to the matrix equation
Ax=b.
Then solve the system and write the solution as a vector.
A=
143
220
324
,
b=
2
14
24
Write the augmented matrix for the linear system that corresponds to the matrix equation
Ax=b.
1432
22014
32424
Solve the system and write the solution as a vector.
x=
x1
x2
x3
=
10
3
0
1.4.22
Let
v1=
0
0
4
,
v2=
0
3
8
,
and
v3=
5
2
12
.
Does
v1,v2,v3
span
3?
Why or why not?
Choose the correct answer below.
Yes. Any vector in
3
except the zero vector can be written as a linear combination of these three vectors.
No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only two rows.
Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row.
No. The set of given vectors spans a plane in
3.
Any of the three vectors can be written as a linear combination of the other two.
1.4.22
Let
v1=
0
0
2
,
v2=
0
6
6
,
and
v3=
6
4
4
.
Does
v1,v2,v3
span
3?
Why or why not?
Choose the correct answer below.
No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only two rows.
No. The set of given vectors spans a plane in
3.
Any of the three vectors can be written as a linear combination of the other two.
Yes. Any vector in
3
except the zero vector can be written as a linear combination of these three vectors.
Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row.
1.4.22
Let
v1=
0
0
4
,
v2=
0
4
8
,
and
v3=
4
3
12
.
Does
v1,v2,v3
span
3?
Why or why not?
Choose the correct answer below.
No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only two rows.
No. The set of given vectors spans a plane in
3.
Any of the three vectors can be written as a linear combination of the other two.
Yes. Any vector in
3
except the zero vector can be written as a linear combination of these three vectors.
Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row.
1.4.22
Let
v1=
0
0
2
,
v2=
0
3
6
,
and
v3=
6
2
4
.
Does
v1,v2,v3
span
3?
Why or why not?
Choose the correct answer below.
No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only two rows.
Yes. Any vector in
3
except the zero vector can be written as a linear combination of these three vectors.
Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row.
No. The set of given vectors spans a plane in
3.
Any of the three vectors can be written as a linear combination of the other two.
1.4.22
Let
v1=
0
0
5
,
v2=
0
2
10
,
and
v3=
3
5
15
.
Does
v1,v2,v3
span
3?
Why or why not?
Choose the correct answer below.
No. The set of given vectors spans a plane in
3.
Any of the three vectors can be written as a linear combination of the other two.
Yes. Any vector in
3
except the zero vector can be written as a linear combination of these three vectors.
No. When the given vectors are written as the columns of a matrix A, A has a pivot position in only two rows.
Yes. When the given vectors are written as the columns of a matrix A, A has a pivot position in every row.
1.4.26
Determine whether the statement below is true or false. Justify the answer.
A vector b is a linear combination of the columns of a matrix A if and only if the equation
Ax=b
has at least one solution.
Choose the correct answer below.
This statement is true. The equation
Ax=b
has the same solution set as the equation
x1a1+x2a2++xnan=b.
This statement is false. If the matrix A is the identity matrix, then the equation
Ax=b
has at least one solution, but b is not a linear combination of the columns of A.
This statement is false. If the equation
Ax=b
has infinitely many solutions, then the vector b cannot be a linear combination of the columns of A.
This statement is true. The equation
Ax=b
is unrelated to whether the vector b is a linear combination of the columns of a matrix A.
1.4.26
Determine whether the statement below is true or false. Justify the answer.
A vector b is a linear combination of the columns of a matrix A if and only if the equation
Ax=b
has at least one solution.
Choose the correct answer below.
This statement is true. The equation
Ax=b
has the same solution set as the equation
x1a1+x2a2++xnan=b.
This statement is false. If the equation
Ax=b
has infinitely many solutions, then the vector b cannot be a linear combination of the columns of A.
This statement is true. The equation
Ax=b
is unrelated to whether the vector b is a linear combination of the columns of a matrix A.
This statement is false. If the matrix A is the identity matrix, then the equation
Ax=b
has at least one solution, but b is not a linear combination of the columns of A.
1.4.27
Determine whether the statement below is true or false. Justify the answer.
The equation
Ax=b
is consistent if the augmented matrix
Ab
has a pivot position in every row.
Choose the correct answer below.
This statement is true. The pivot positions in the augmented matrix
Ab
always occur in the columns that represent A.
This statement is false. If the augmented matrix
Ab
has a pivot position in every row, the equation
Ax=b
may or may not be consistent. One pivot position may be in the column representing
b.
This statement is false. The augmented matrix
Ab
cannot have a pivot position in every row because it has more columns than rows.
This statement is true. If the augmented matrix
Ab
has a pivot position in every row, then the equation
Ax=b
has a solution for each b in
m.
1.4.33
Determine whether the statement below is true or false. Justify the answer.
If A is an
m×n
matrix and if the equation
Ax=b
is inconsistent for some b in
m,
then A cannot have a pivot position in every row.
Choose the correct answer below.
This statement is true. If A is an
m×n
matrix and if the equation
Ax=b
is inconsistent for some b in
m,
then the columns of A span
m.
This statement is false. If A is an
m×n
matrix and if the equation
Ax=b
is inconsistent for some b in
m,
then the equation
Ax=b
has a solution for each b in
m.
This statement is false. Since the equation
Ax=b
has a solution for each b in
m,
the equation
Ax=b
is consistent for each b in
m.
This statement is true. If A is an
m×n
matrix and if the equation
Ax=b
is inconsistent for some b in
m,
then the equation
Ax=b
has no solution for some b in
m.
1.4.33
Determine whether the statement below is true or false. Justify the answer.
If A is an
m×n
matrix and if the equation
Ax=b
is inconsistent for some b in
m,
then A cannot have a pivot position in every row.
Choose the correct answer below.
This statement is true. If A is an
m×n
matrix and if the equation
Ax=b
is inconsistent for some b in
m,
then the equation
Ax=b
has no solution for some b in
m.
This statement is false. If A is an
m×n
matrix and if the equation
Ax=b
is inconsistent for some b in
m,
then the equation
Ax=b
has a solution for each b in
m.
This statement is true. If A is an
m×n
matrix and if the equation
Ax=b
is inconsistent for some b in
m,
then the columns of A span
m.
This statement is false. Since the equation
Ax=b
has a solution for each b in
m,
the equation
Ax=b
is consistent for each b in
m.
1.4.41
Let A be a
3×2
matrix. Explain why the equation
Ax=b
cannot be consistent for all b in
3.
Generalize your argument to the case of an arbitrary A with more rows than columns.
Why is the equation
Ax=b
not consistent for all b in
3?
When written in reduced echelon form, any
3×2
matrix will have at least one row of all zeros. When solving
Ax=b,
that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side.
When written in reduced echelon form, any
3×2
matrix will have at least one column of all zeros. Since there is not a pivot in every column, the matrix cannot be consistent.
When written in reduced echelon form, any
3×2
matrix will have a pivot in every row. Because there are more rows than columns, this is too many pivots, and the system will be inconsistent.
When written in reduced echelon form, any
3×2
matrix will have at least one row of all zeros. When solving
Ax=b,
that row will represent an equation with infinitely many solutions.
Let A be an
m×n
matrix, where
m>n.
Why is
Ax=b
not consistent for all b in
m?
When written in reduced echelon form, any
m×n
matrix will have at least one column of all zeros. Since there is not a pivot in every column, the matrix cannot be consistent.
When written in reduced echelon form, any
m×n
matrix will have at least one row of all zeros. When solving
Ax=b,
that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side.
If A has more rows than columns, the number of pivots cannot be determined without knowing more information about A, and so A cannot be consistent.
When written in reduced echelon form, any
m×n
matrix will have at least one row of all zeros. When solving
Ax=b,
that row will represent an equation with infinitely many solutions.
When written in reduced echelon form, any
m×n
matrix will have a pivot in every row. Because there are more rows than columns, this is too many pivots, and the system will be inconsistent.
1.4.41
Let A be a
3×2
matrix. Explain why the equation
Ax=b
cannot be consistent for all b in
3.
Generalize your argument to the case of an arbitrary A with more rows than columns.
Why is the equation
Ax=b
not consistent for all b in
3?
When written in reduced echelon form, any
3×2
matrix will have at least one column of all zeros. Since there is not a pivot in every column, the matrix cannot be consistent.
When written in reduced echelon form, any
3×2
matrix will have at least one row of all zeros. When solving
Ax=b,
that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side.
When written in reduced echelon form, any
3×2
matrix will have at least one row of all zeros. When solving
Ax=b,
that row will represent an equation with infinitely many solutions.
When written in reduced echelon form, any
3×2
matrix will have a pivot in every row. Because there are more rows than columns, this is too many pivots, and the system will be inconsistent.
Let A be an
m×n
matrix, where
m>n.
Why is
Ax=b
not consistent for all b in
m?
When written in reduced echelon form, any
m×n
matrix will have at least one row of all zeros. When solving
Ax=b,
that row will represent an equation with infinitely many solutions.
When written in reduced echelon form, any
m×n
matrix will have a pivot in every row. Because there are more rows than columns, this is too many pivots, and the system will be inconsistent.
When written in reduced echelon form, any
m×n
matrix will have at least one row of all zeros. When solving
Ax=b,
that row will represent an equation with a zero on the left side and a possibly nonzero entry of b on the right side.
When written in reduced echelon form, any
m×n
matrix will have at least one column of all zeros. Since there is not a pivot in every column, the matrix cannot be consistent.
If A has more rows than columns, the number of pivots cannot be determined without knowing more information about A, and so A cannot be consistent.
1.4.44
Suppose A is a
3×3
matrix and b is a vector in
3
with the property what
Ax=b
has a unique solution. Explain why the columns of A must span
3.
Choose the correct answer below.
The equation has a unique solution so for each pair of vectors x and b there is only one possible matrix A. Therefore the columns of A must span
3.
Matrix A is a square matrix, so when computing
Ax,
the row-vector rule shows that the columns of A must span
3.
When b is written as a linear combination of the columns of A, it simplifies to the vector of weights,
x.
Therefore the columns of A must span
3.
The reduced echelon form of A must have a pivot in each row or there would be more than one possible solution for the equation
Ax=b.
Therefore the columns of A must span
3.
1.4.44
Suppose A is a
3×3
matrix and b is a vector in
3
with the property what
Ax=b
has a unique solution. Explain why the columns of A must span
3.
Choose the correct answer below.
Matrix A is a square matrix, so when computing
Ax,
the row-vector rule shows that the columns of A must span
3.
The equation has a unique solution so for each pair of vectors x and b there is only one possible matrix A. Therefore the columns of A must span
3.
When b is written as a linear combination of the columns of A, it simplifies to the vector of weights,
x.
Therefore the columns of A must span
3.
The reduced echelon form of A must have a pivot in each row or there would be more than one possible solution for the equation
Ax=b.
Therefore the columns of A must span
3.
1.5.1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
2x19x2+11x3
=
0
2x111x2+4x3
=
0
4x1+2x2+7x3
=
0
Choose the correct answer below.
It is impossible to determine.
The system has a nontrivial solution.
The system has only a trivial solution.
1.5.1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
2x19x2+11x3
=
0
2x111x2+4x3
=
0
4x1+2x2+7x3
=
0
Choose the correct answer below.
It is impossible to determine.
The system has a nontrivial solution.
The system has only a trivial solution.
1.5.1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
6x13x2+15x3
=
0
6x19x26x3
=
0
12x1+6x2+21x3
=
0
Choose the correct answer below.
The system has a nontrivial solution.
The system has only a trivial solution.
It is impossible to determine.
1.5.1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
2x15x2+8x3
=
0
2x17x2+x3
=
0
4x1+2x2+7x3
=
0
Choose the correct answer below.
The system has only a trivial solution.
It is impossible to determine.
The system has a nontrivial solution.
1.5.1
Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
4x12x2+10x3
=
0
4x12x27x3
=
0
8x1+4x2+14x3
=
0
Choose the correct answer below.
The system has a nontrivial solution.
It is impossible to determine.
The system has only a trivial solution.
1.5.2
Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
x1
3x2
+
6x3
=
0
3x1
+
4x2
10x3
=
0
x1
+
2x2
+
10x3
=
0
Choose the correct answer below.
The system has a nontrivial solution.
The system has only a trivial solution.
It is impossible to determine.
1.5.2
Determine if the system has a nontrivial solution. Try to use as few row operations as possible.
x1
2x2
+
6x3
=
0
4x1
+
3x2
15x3
=
0
x1
+
3x2
+
10x3
=
0
Choose the correct answer below.
The system has a nontrivial solution.
The system has only a trivial solution.
It is impossible to determine.
1.5.5
Write the solution set of the given homogeneous system in parametric vector form.
3x1+3x2+6x3
=
0
where the solution set is
x=
x1
x2
x3
6x16x212x3
=
0
7x2+14x3
=
0
x=x3
4
2
1
(Type an integer or simplified fraction for each matrix element.)
1.5.5
Write the solution set of the given homogeneous system in parametric vector form.
2x1+2x2+4x3
=
0
where the solution set is
x=
x1
x2
x3
4x14x28x3
=
0
6x2+18x3
=
0
x=x3
5
3
1
(Type an integer or simplified fraction for each matrix element.)
1.5.7
Describe all solutions of
Ax=0
in parametric vector form, where A is row equivalent to the given matrix.
1458
0155
x=x3
15
5
1
0
+x4
12
5
0
1
(Type an integer or fraction for each matrix element.)
1.5.7
Describe all solutions of
Ax=0
in parametric vector form, where A is row equivalent to the given matrix.
1355
0128
x=x3
1
2
1
0
+x4
19
8
0
1
(Type an integer or fraction for each matrix element.)
1.5.8
Describe all solutions of
Ax=0
in parametric vector form, where A is row equivalent to the given matrix.
1494
0142
x=x3
7
4
1
0
+x4
4
2
0
1
(Type an integer or fraction for each matrix element.)
1.5.8
Describe all solutions of
Ax=0
in parametric vector form, where A is row equivalent to the given matrix.
1253
0134
x=x3
1
3
1
0
+x4
5
4
0
1
(Type an integer or fraction for each matrix element.)
1.5.8
Describe all solutions of
Ax=0
in parametric vector form, where A is row equivalent to the given matrix.
1272
0152
x=x3
3
5
1
0
+x4
2
2
0
1
(Type an integer or fraction for each matrix element.)
1.5.8
Describe all solutions of
Ax=0
in parametric vector form, where A is row equivalent to the given matrix.
1475
0127
x=x3
1
2
1
0
+x4
23
7
0
1
(Type an integer or fraction for each matrix element.)
1.5.9
Describe all solutions of
Ax=0
in parametric vector form, where A is row equivalent to the given matrix.
4168
142
x=x2
4
1
0
+x3
2
0
1
(Type an integer or fraction for each matrix element.)
1.5.19
Describe the solutions of the first system of equations below in parametric vector form. Provide a geometric comparison with the solution set of the second system of equations below.
3x1+3x2+6x3
=
12
3x1+3x2+6x3
=
0
9x19x218x3
=
36
9x19x218x3
=
0
4x24x3
=
16
4x24x3
=
0
Describe the solution set,
x=
x1
x2
x3
,
of the first system of equations in parametric vector form. Select the correct choice below and fill in the answer box(es) within your choice.
(Type an integer or fraction for each matrix element.)
x=
x=x2
x=
8
4
0
+x3
1
1
1
x=x2+x3
Which option best compares the two systems?
The solution set of the first system is a plane parallel to the line that is the solution set of the second system.
The solution set of the first system is a plane parallel to the plane that is the solution set of the second system.
The solution set of the first system is a line parallel to the line that is the solution set of the second system.
The solution set of the first system is a line perpendicular to the line that is the solution set of the second system.
1.5.19
Describe the solutions of the first system of equations below in parametric vector form. Provide a geometric comparison with the solution set of the second system of equations below.
2x1+2x2+4x3
=
8
2x1+2x2+4x3
=
0
6x16x212x3
=
24
6x16x212x3
=
0
7x2+14x3
=
14
7x2+14x3
=
0
Describe the solution set,
x=
x1
x2
x3
,
of the first system of equations in parametric vector form. Select the correct choice below and fill in the answer box(es) within your choice.
(Type an integer or fraction for each matrix element.)
x=
x=x2
x=
6
2
0
+x3
4
2
1
x=x2+x3
Which option best compares the two systems?
The solution set of the first system is a line parallel to the line that is the solution set of the second system.
The solution set of the first system is a line perpendicular to the line that is the solution set of the second system.
The solution set of the first system is a plane parallel to the line that is the solution set of the second system.
The solution set of the first system is a plane parallel to the plane that is the solution set of the second system.
1.5.19
Describe the solutions of the first system of equations below in parametric vector form. Provide a geometric comparison with the solution set of the second system of equations below.
2x1+2x2+4x3
=
8
2x1+2x2+4x3
=
0
4x14x28x3
=
16
4x14x28x3
=
0
3x2+6x3
=
15
3x2+6x3
=
0
Describe the solution set,
x=
x1
x2
x3
,
of the first system of equations in parametric vector form. Select the correct choice below and fill in the answer box(es) within your choice.
(Type an integer or fraction for each matrix element.)
x=
x=x2
x=
9
5
0
+x3
4
2
1
x=x2+x3
Which option best compares the two systems?
The solution set of the first system is a line parallel to the line that is the solution set of the second system.
The solution set of the first system is a line perpendicular to the line that is the solution set of the second system.
The solution set of the first system is a plane parallel to the plane that is the solution set of the second system.
The solution set of the first system is a plane parallel to the line that is the solution set of the second system.
1.5.19
Describe the solutions of the first system of equations below in parametric vector form. Provide a geometric comparison with the solution set of the second system of equations below.
2x1+2x2+4x3
=
8
2x1+2x2+4x3
=
0
4x14x28x3
=
16
4x14x28x3
=
0
3x2+6x3
=
15
3x2+6x3
=
0
Describe the solution set,
x=
x1
x2
x3
,
of the first system of equations in parametric vector form. Select the correct choice below and fill in the answer box(es) within your choice.
(Type an integer or fraction for each matrix element.)
x=
x=x2
x=
9
5
0
+x3
4
2
1
x=x2+x3
Which option best compares the two systems?
The solution set of the first system is a plane parallel to the plane that is the solution set of the second system.
The solution set of the first system is a line perpendicular to the line that is the solution set of the second system.
The solution set of the first system is a line parallel to the line that is the solution set of the second system.
The solution set of the first system is a plane parallel to the line that is the solution set of the second system.
1.5.28
Determine whether the statement below is true or false. Justify the answer.
If x is a nontrivial solution of
Ax=0,
then every entry in x is nonzero.
Choose the correct answer below.
The statement is false. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero.
The statement is true. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x cannot have any zero entries.
The statement is true. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero.
The statement is false. The only solution of
Ax=0
is the zero vector. Thus, a nontrivial solution does not exist.
1.5.28
Determine whether the statement below is true or false. Justify the answer.
If x is a nontrivial solution of
Ax=0,
then every entry in x is nonzero.
Choose the correct answer below.
The statement is false. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero.
The statement is true. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero.
The statement is false. The only solution of
Ax=0
is the zero vector. Thus, a nontrivial solution does not exist.
The statement is true. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x cannot have any zero entries.
1.5.28
Determine whether the statement below is true or false. Justify the answer.
If x is a nontrivial solution of
Ax=0,
then every entry in x is nonzero.
Choose the correct answer below.
The statement is true. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x cannot have any zero entries.
The statement is false. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero.
The statement is true. A nontrivial solution of
Ax=0
is a nonzero vector x that satisfies
Ax=0.
Thus, a nontrivial solution x can have some zero entries so long as not all of its entries are zero.
The statement is false. The only solution of
Ax=0
is the zero vector. Thus, a nontrivial solution does not exist.
1.5.31
Determine whether the statement below is true or false. Justify the answer.
The homogenous equation
Ax=0
has the trivial solution if and only if the equation has at least one free variable.
Choose the correct answer below.
The statement is false. The homogeneous equation
Ax=0
always has the trivial solution.
The statement is true. The homogenous equation
Ax=0
has the trivial solution if and only if the equation has at least one free variable which implies that the equation has a nontrivial solution.
The statement is false. The homogeneous equation
Ax=0
never has the trivial solution.
The statement is true. The homogeneous equation
Ax=0
has the trivial solution if and only if the matrix A has a row of zeros which implies the equation has at least one free variable.
1.5.31
Determine whether the statement below is true or false. Justify the answer.
The homogenous equation
Ax=0
has the trivial solution if and only if the equation has at least one free variable.
Choose the correct answer below.
The statement is true. The homogenous equation
Ax=0
has the trivial solution if and only if the equation has at least one free variable which implies that the equation has a nontrivial solution.
The statement is false. The homogeneous equation
Ax=0
never has the trivial solution.
The statement is true. The homogeneous equation
Ax=0
has the trivial solution if and only if the matrix A has a row of zeros which implies the equation has at least one free variable.
The statement is false. The homogeneous equation
Ax=0
always has the trivial solution.
1.5.35
Determine whether the statement below is true or false. Justify the answer.
The solution set of
Ax=b
is the set of all vectors of the form
w=p+vh,
where
vh
is any solution of the equation
Ax=0.
Choose the correct answer below.
The statement is false. The solution set could be empty. The statement is only true when the equation
Ax=b
is consistent for some given
b,
and there exists a vector p such that p is a solution.
The statement is false. The solution set could be empty. The statement is only true when the equation
Ax=b
is inconsistent for some given
b,
and there exists a vector p such that p is a solution.
The statement is false. The solution set could be the trivial solution. The statement is only true when the equation
Ax=b
is inconsistent for some given
b,
and there exists a vector p such that p is a solution.
The statement is true. The equation
Ax=b
is always consistent and there always exists a vector p that is a solution.
1.5.35
Determine whether the statement below is true or false. Justify the answer.
The solution set of
Ax=b
is the set of all vectors of the form
w=p+vh,
where
vh
is any solution of the equation
Ax=0.
Choose the correct answer below.
The statement is true. The equation
Ax=b
is always consistent and there always exists a vector p that is a solution.
The statement is false. The solution set could be the trivial solution. The statement is only true when the equation
Ax=b
is inconsistent for some given
b,
and there exists a vector p such that p is a solution.
The statement is false. The solution set could be empty. The statement is only true when the equation
Ax=b
is consistent for some given
b,
and there exists a vector p such that p is a solution.
The statement is false. The solution set could be empty. The statement is only true when the equation
Ax=b
is inconsistent for some given
b,
and there exists a vector p such that p is a solution.
1.5.40
If
b0,
can the solution set of
Ax=b
be a plane through the origin?
Choose the correct answer.
Yes. Since the solution set of
Ax=0
contains the origin, the solution set of
Ax=b
must contain the origin.
Yes. The solution set of
Ax=b
is always represented as a plane through the origin.
No. The solution set of
Ax=b
contains the origin if and only if
Ax=b
is inconsistent, which is not true for any particular vector
b.
No. If the solution set of
Ax=b
contained the origin, then 0 would satisfy
A0=b,
which is not true since b is not the zero vector.
1.5.40
If
b0,
can the solution set of
Ax=b
be a plane through the origin?
Choose the correct answer.
No. The solution set of
Ax=b
contains the origin if and only if
Ax=b
is inconsistent, which is not true for any particular vector
b.
No. If the solution set of
Ax=b
contained the origin, then 0 would satisfy
A0=b,
which is not true since b is not the zero vector.
Yes. The solution set of
Ax=b
is always represented as a plane through the origin.
Yes. Since the solution set of
Ax=0
contains the origin, the solution set of
Ax=b
must contain the origin.
1.5.44
A is a
2×5
matrix with two pivot positions.
(a) Does the equation
Ax=0
have a nontrivial solution?
(b) Does the equation
Ax=b
have at least one solution for every possible
b?
(a) Does the equation
Ax=0
have a nontrivial solution?
Yes
No
(b) Does the equation
Ax=b
have at least one solution for every possible
b?
No
Yes
1.5.44
A is a
2×5
matrix with two pivot positions.
(a) Does the equation
Ax=0
have a nontrivial solution?
(b) Does the equation
Ax=b
have at least one solution for every possible
b?
(a) Does the equation
Ax=0
have a nontrivial solution?
Yes
No
(b) Does the equation
Ax=b
have at least one solution for every possible
b?
Yes
No
1.6.1
Suppose an economy has only two sectors: Goods and Services. Each year, Goods sells
75%
of its outputs to Services and keeps the rest, while Services sells
66%
of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures.
0.34
0.75
0.66
0.25
GoodsServices
Denote the prices (that is, dollar values) of the total annual outputs of the Goods and Services sectors by
pG
and
pS,
respectively.
If
pS=$1000,
then
pG=$880.
(Type an integer or a decimal.)
If
pS=$75,
then
pG=$66.
(Type an integer or a decimal.)
1.6.1
Suppose an economy has only two sectors: Goods and Services. Each year, Goods sells
80%
of its outputs to Services and keeps the rest, while Services sells
65%
of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures.
0.35
0.8
0.65
0.2
GoodsServices
Denote the prices (that is, dollar values) of the total annual outputs of the Goods and Services sectors by
pG
and
pS,
respectively.
If
pS=$1000,
then
pG=$812.5.
(Type an integer or a decimal.)
If
pS=$80,
then
pG=$65.
(Type an integer or a decimal.)
1.6.1
Suppose an economy has only two sectors: Goods and Services. Each year, Goods sells
75%
of its outputs to Services and keeps the rest, while Services sells
60%
of its output to Goods and retains the rest. Find equilibrium prices for the annual outputs of the Goods and Services sectors that make each sector's income match its expenditures.
0.4
0.75
0.6
0.25
GoodsServices
Denote the prices (that is, dollar values) of the total annual outputs of the Goods and Services sectors by
pG
and
pS,
respectively.
If
pS=$1000,
then
pG=$800.
(Type an integer or a decimal.)
If
pS=$75,
then
pG=$60.
(Type an integer or a decimal.)
1.6.2
Suppose an economy consists of the Coal, Electric, and Steel sectors. Denote the prices (that is, dollar values) of the total annual outputs of the Coal, Electric, and Steel sectors by
pC,
pE,
and
pS,
respectively. Suppose the general solution to find equilibrium prices that make each sector's income match its expenditures is
pC=0.91pS,
pE=0.86pS,
and
pS
is free. One set of equilibrium prices for this economy is
pC=$91,
pE=$86,
and
pS=$100.
Find another set. Suppose the same economy used Japanese yen instead of dollars to measure the values of the various sector's output. Would this change the problem in any way? Discuss.
If
pS=$200,
then
pC=$182
and
pE=$172.
(Type integers or decimals.)
How would changing the unit of measurement to Japanese yen change this problem?
It has the same effect as multiplying all equilibrium prices by a constant. The ratios of the prices will change, but the prices will remain the same.
It has the same effect as multiplying all equilibrium prices by a constant. The prices will change, but the ratios of the prices will remain the same.
It has the same effect as multiplying all equilibrium prices by their reciprocal and some constant. The prices will change, but the ratios of the prices will remain the same.
It has the same effect as multiplying all equilibrium prices by their reciprocal and some constant. The prices and the ratios of the prices will change.
1.6.2
Suppose an economy consists of the Coal, Electric, and Steel sectors. Denote the prices (that is, dollar values) of the total annual outputs of the Coal, Electric, and Steel sectors by
pC,
pE,
and
pS,
respectively. Suppose the general solution to find equilibrium prices that make each sector's income match its expenditures is
pC=0.87pS,
pE=0.89pS,
and
pS
is free. One set of equilibrium prices for this economy is
pC=$87,
pE=$89,
and
pS=$100.
Find another set. Suppose the same economy used Japanese yen instead of dollars to measure the values of the various sector's output. Would this change the problem in any way? Discuss.
If
pS=$200,
then
pC=$174
and
pE=$178.
(Type integers or decimals.)
How would changing the unit of measurement to Japanese yen change this problem?
It has the same effect as multiplying all equilibrium prices by their reciprocal and some constant. The prices will change, but the ratios of the prices will remain the same.
It has the same effect as multiplying all equilibrium prices by a constant. The prices will change, but the ratios of the prices will remain the same.
It has the same effect as multiplying all equilibrium prices by their reciprocal and some constant. The prices and the ratios of the prices will change.
It has the same effect as multiplying all equilibrium prices by a constant. The ratios of the prices will change, but the prices will remain the same.
1.6.3
Consider an economy with three sectors, Chemicals & Metals, Fuels & Power, and Machinery. Chemicals sells
30%
of its output to Fuels and
60%
to Machinery and retains the rest. Fuels sells
80%
of its output to Chemicals and
10%
to Machinery and retains the rest. Machinery sells
30%
of its output to Chemicals and
20%
to Fuels and retains the rest. Complete parts (a) through (c) below.
a. Construct the exchange table for this economy.
Distribution of Output from:
Chemicals
Fuels
Machinery
Purchased by:
0.10
0.80
0.30
Chemicals
0.30
0.10
0.20
Fuels
0.60
0.10
0.50
Machinery
(Type integers or decimals.)
b. Develop a system of equations that leads to prices at which each sector's income matches its expenses. Then write the augmented matrix that can be row reduced to find these prices. The first, second, and third columns of the matrix should correspond to Chemicals, Fuels, and Machinery, respectively.
The augmented matrix is
0.900.800.300
0.300.900.200
0.600.100.500
.
(Type an integer or decimal for each matrix element.)
c. Find a set of equilibrium prices when the price for the Machinery output is
20
units.
pChemicals=15.1,
pFuels=9.5,
pMachinery=20
(Type integers or decimals rounded to the nearest tenth as needed.)
1.6.3
Consider an economy with three sectors, Chemicals & Metals, Fuels & Power, and Machinery. Chemicals sells
20%
of its output to Fuels and
60%
to Machinery and retains the rest. Fuels sells
70%
of its output to Chemicals and
20%
to Machinery and retains the rest. Machinery sells
50%
of its output to Chemicals and
30%
to Fuels and retains the rest. Complete parts (a) through (c) below.
a. Construct the exchange table for this economy.
Distribution of Output from:
Chemicals
Fuels
Machinery
Purchased by:
0.20
0.70
0.50
Chemicals
0.20
0.10
0.30
Fuels
0.60
0.20
0.20
Machinery
(Type integers or decimals.)
b. Develop a system of equations that leads to prices at which each sector's income matches its expenses. Then write the augmented matrix that can be row reduced to find these prices. The first, second, and third columns of the matrix should correspond to Chemicals, Fuels, and Machinery, respectively.
The augmented matrix is
0.80.70.50
0.20.90.30
0.60.20.80
.
(Type an integer or decimal for each matrix element.)
c. Find a set of equilibrium prices when the price for the Machinery output is
90
units.
pChemicals=102.4,
pFuels=52.8,
pMachinery=90.0
(Type integers or decimals rounded to the nearest tenth as needed.)
1.6.4
Suppose an economy has four sectors: Mining, Lumber, Energy, and Transportation. Mining sells
15%
of its output to Lumber,
70%
to Energy, and retains the rest. Lumber sells
15%
of its output to Mining,
50%
to Energy,
20%
to Transportation, and retains the rest. Energy sells
30%
of its output to Mining,
10%
to Lumber,
25%
to Transportation, and retains the rest. Transportation sells
10%
of its output to Mining,
20%
to Lumber,
50%
to Energy, and retains the rest.
a. Construct the exchange table for this economy.
b. Find a set of equilibrium prices for this economy.
a. Complete the exchange table below.
Distribution of Output from:
Mining
Lumber
Energy
Transportation
Purchased by:
0.15
0.15
0.30
0.10
Mining
0.15
0.15
0.10
0.20
Lumber
0.70
0.50
0.35
0.50
Energy
0.00
0.20
0.25
0.20
Transportation
(Type integers or decimals.)
b. Denote the prices (that is, dollar values) of the total annual outputs of the Mining, Lumber, Energy, and Transportation sectors by
pM,
pL,
pE,
and
pT,
respectively.
If
pT=$100,
then
pM=$117,
pL=$75,
and
pE=$260.
(Round to the nearest dollar as needed.)
1.6.4
Suppose an economy has four sectors: Mining, Lumber, Energy, and Transportation. Mining sells
10%
of its output to Lumber,
50%
to Energy, and retains the rest. Lumber sells
10%
of its output to Mining,
40%
to Energy,
25%
to Transportation, and retains the rest. Energy sells
30%
of its output to Mining,
10%
to Lumber,
20%
to Transportation, and retains the rest. Transportation sells
25%
of its output to Mining,
10%
to Lumber,
50%
to Energy, and retains the rest.
a. Construct the exchange table for this economy.
b. Find a set of equilibrium prices for this economy.
a. Complete the exchange table below.
Distribution of Output from:
Mining
Lumber
Energy
Transportation
Purchased by:
0.40
0.10
0.30
0.25
Mining
0.10
0.25
0.10
0.10
Lumber
0.50
0.40
0.40
0.50
Energy
0.00
0.25
0.20
0.15
Transportation
(Type integers or decimals.)
b. Denote the prices (that is, dollar values) of the total annual outputs of the Mining, Lumber, Energy, and Transportation sectors by
pM,
pL,
pE,
and
pT,
respectively.
If
pT=$100,
then
pM=$215,
pL=$85,
and
pE=$319.
(Round to the nearest dollar as needed.)
1.6.5
Balance the chemical equation.
CS2+NH3H2S+NH4SCN
Assume the coefficient of
NH4SCN
is
1.
What is the balanced equation?
1CS2+2NH31H2S+1NH4SCN
1.6.5
Balance the chemical equation.
B2S3+H2OH3BO3+H2S
Assume the coefficient of
H2S
is
3.
What is the balanced equation?
1B2S3+6H2O2H3BO3+3H2S
1.6.5
Balance the chemical equation.
B2S3+H2OH3BO3+H2S
Assume the coefficient of
H2S
is
3.
What is the balanced equation?
1B2S3+6H2O2H3BO3+3H2S
1.6.5
Balance the chemical equation.
B2S3+H2OH3BO3+H2S
Assume the coefficient of
H2S
is
3.
What is the balanced equation?
1B2S3+6H2O2H3BO3+3H2S
1.6.7
Balance the following chemical equation.
NaHCO3+H3C6H5O7Na3C6H5O7+H2O+CO2
Assume the coefficient of
CO2
is
3.
What is the balanced equation?
3NaHCO3+1H3C6H5O71Na3C6H5O7+3H2O+3CO2
1.6.9
Balance the chemical equation.
PbN6+CrMn2O8Pb3O4+Cr2O3+MnO2+NO
Assume the coefficient of
NO
is
90.
What is the balanced equation?
15PbN6+44CrMn2O85Pb3O4+22Cr2O3+88MnO2+90NO
1.6.9
Balance the chemical equation.
Au+HCl+HNO3AuCl3+NO+H2O
Assume the coefficient of
H2O
is
2.
What is the balanced equation?
1Au+3HCl+1HNO31AuCl3+1NO+2H2O
1.6.9
Balance the chemical equation.
PbN6+CrMn2O8Pb3O4+Cr2O3+MnO2+NO
Assume the coefficient of
NO
is
90.
What is the balanced equation?
15PbN6+44CrMn2O85Pb3O4+22Cr2O3+88MnO2+90NO
1.6.9
Balance the chemical equation.
Au+HCl+HNO3AuCl3+NO+H2O
Assume the coefficient of
H2O
is
2.
What is the balanced equation?
1Au+3HCl+1HNO31AuCl3+1NO+2H2O
1.7.5
Determine if the columns of the matrix form a linearly independent set.
Justify your answer.
0816
3114
154
152
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
If A is the given matrix, then the augmented matrix
08160
31140
1540
1520
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A do not form a linearly independent set.
1.7.5
Determine if the columns of the matrix form a linearly independent set.
Justify your answer.
039
217
148
142
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
0390
2170
1480
1420
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A do not form a linearly independent set.
1.7.5
Determine if the columns of the matrix form a linearly independent set.
Justify your answer.
0816
3114
156
152
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
If A is the given matrix, then the augmented matrix
08160
31140
1560
1520
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A do not form a linearly independent set.
1.7.5
Determine if the columns of the matrix form a linearly independent set.
Justify your answer.
039
217
146
142
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
If A is the given matrix, then the augmented matrix
0390
2170
1460
1420
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A form a linearly independent set.
1.7.5
Determine if the columns of the matrix form a linearly independent set.
Justify your answer.
039
217
147
142
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
0390
2170
1470
1420
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A form a linearly independent set.
1.7.6
Determine if the columns of the matrix form a linearly independent set.
Justify your answer.
430
016
116
2112
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
4300
0160
1160
21120
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A form a linearly independent set.
1.7.6
Determine if the columns of the matrix form a linearly independent set.
Justify your answer.
430
015
115
2110
Select the correct choice below and fill in the answer box within your choice.
(Type an integer or simplified fraction for each matrix element.)
If A is the given matrix, then the augmented matrix
4300
0150
1150
21100
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has more than one solution. Therefore, the columns of A do not form a linearly independent set.
If A is the given matrix, then the augmented matrix
represents the equation
Ax=0.
The reduced echelon form of this matrix indicates that
Ax=0
has only the trivial solution. Therefore, the columns of A do not form a linearly independent set.
1.7.8
Determine if the columns of the matrix form a linearly independent set. Justify your answer.
123
243
Choose the correct answer below.
The columns of the matrix do not form a linearly independent set because the set contains more vectors than there are entries in each vector.
The columns of the matrix do form a linearly independent set because there are more entries in each vector than there are vectors in the set.
The columns of the matrix do form a linearly independent set because the set contains more vectors than there are entries in each vector.
The columns of the matrix do not form a linearly independent set because there are more entries in each vector than there are vectors in the set.
1.7.8
Determine if the columns of the matrix form a linearly independent set. Justify your answer.
1234
2464
0115
Choose the correct answer below.
The columns of the matrix do not form a linearly independent set because the set contains more vectors than there are entries in each vector.
The columns of the matrix do not form a linearly independent set because there are more entries in each vector than there are vectors in the set.
The columns of the matrix do form a linearly independent set because the set contains more vectors than there are entries in each vector.
The columns of the matrix do form a linearly independent set because there are more entries in each vector than there are vectors in the set.
1.7.13
Find the value(s) of h for which the vectors are linearly dependent.
1
3
2
,
4
11
8
,
3
h
6
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The vectors are linearly dependent if
h=
because the related matrix will have a free variable. (Type an integer or a simplified fraction.)
The vectors are linearly dependent for all values of h because the related matrix always has a free variable.
The vectors are linearly independent for all values of h because the related matrix never has a free variable.
1.7.13
Find the value(s) of h for which the vectors are linearly dependent.
1
5
2
,
2
9
4
,
2
h
4
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The vectors are linearly dependent if
h=
because the related matrix will have a free variable. (Type an integer or a simplified fraction.)
The vectors are linearly dependent for all values of h because the related matrix always has a free variable.
The vectors are linearly independent for all values of h because the related matrix never has a free variable.
1.7.14
Find the value(s) of h for which the vectors are linearly dependent. Justify your answer.
1
3
6
,
4
13
8
,
4
1
h
The value(s) of h which makes the vectors linearly dependent is(are)
232
because this will cause
x3
to be a
free
variable.
(Use a comma to separate answers as needed.)
1.7.14
Find the value(s) of h for which the vectors are linearly dependent. Justify your answer.
1
2
4
,
3
7
6
,
3
1
h
The value(s) of h which makes the vectors linearly dependent is(are)
54
because this will cause
x3
to be a
free
variable.
(Use a comma to separate answers as needed.)
1.7.16
Determine by inspection whether the vectors are linearly independent. Justify your answer.
6
12
9
,
4
8
6
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The set of vectors is linearly dependent because
23
times the first vector is equal to the second vector.
(Type an integer or a simplified fraction.)
The set of vectors is linearly dependent because neither vector is the zero vector.
The set of vectors is linearly independent because neither vector is a multiple of the other vector.
1.7.16
Determine by inspection whether the vectors are linearly independent. Justify your answer.
6
4
2
,
9
6
3
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The set of vectors is linearly dependent because
32
times the first vector is equal to the second vector.
(Type an integer or a simplified fraction.)
The set of vectors is linearly dependent because neither vector is the zero vector.
The set of vectors is linearly independent because neither vector is a multiple of the other vector.
1.7.17
Determine by inspection whether the vectors are linearly independent. Justify your answer.
4
2
3
,
0
0
0
,
2
4
4
Choose the correct answer below.
The set of vectors is linearly independent because
times the first vector is equal to the second vector.
(Type an integer or a simplified fraction.)
The set of vectors is linearly dependent because
times the first vector is equal to the third vector.
(Type an integer or a simplified fraction.)
The set of vectors is linearly dependent because one of the vectors is the zero vector.
The set of vectors is linearly independent because none of the vectors are multiples of the other vectors.
1.7.17
Determine by inspection whether the vectors are linearly independent. Justify your answer.
6
2
3
,
0
0
0
,
4
3
2
Choose the correct answer below.
The set of vectors is linearly dependent because
times the first vector is equal to the third vector.
(Type an integer or a simplified fraction.)
The set of vectors is linearly independent because
times the first vector is equal to the second vector.
(Type an integer or a simplified fraction.)
The set of vectors is linearly dependent because one of the vectors is the zero vector.
The set of vectors is linearly independent because none of the vectors are multiples of the other vectors.
1.7.19
Determine by inspection whether the vectors are linearly independent. Justify your answer.
8
16
4
,
2
4
1
Choose the correct answer below.
The set is linearly independent because the first vector is a multiple of the other vector. The entries in the first vector are
4
times the corresponding entry in the second vector.
The set is linearly dependent because neither vector is a multiple of the other vector. Two of the entries in the first vector are
4
times the corresponding entry in the second vector. But this multiple does not work for the third entries.
The set is linearly independent because neither vector is a multiple of the other vector. Two of the entries in the first vector are
4
times the corresponding entry in the second vector. But this multiple does not work for the third entries.
The set is linearly dependent because the first vector is a multiple of the other vector. The entries in the first vector are
4
times the corresponding entry in the second vector.
1.7.19
Determine by inspection whether the vectors are linearly independent. Justify your answer.
10
20
5
,
2
4
1
Choose the correct answer below.
The set is linearly dependent because the first vector is a multiple of the other vector. The entries in the first vector are
5
times the corresponding entry in the second vector.
The set is linearly independent because the first vector is a multiple of the other vector. The entries in the first vector are
5
times the corresponding entry in the second vector.
The set is linearly independent because neither vector is a multiple of the other vector. Two of the entries in the first vector are
5
times the corresponding entry in the second vector. But this multiple does not work for the third entries.
The set is linearly dependent because neither vector is a multiple of the other vector. Two of the entries in the first vector are
5
times the corresponding entry in the second vector. But this multiple does not work for the third entries.
1.7.19
Determine by inspection whether the vectors are linearly independent. Justify your answer.
8
12
4
,
2
3
1
Choose the correct answer below.
The set is linearly independent because neither vector is a multiple of the other vector. Two of the entries in the first vector are
4
times the corresponding entry in the second vector. But this multiple does not work for the third entries.
The set is linearly independent because the first vector is a multiple of the other vector. The entries in the first vector are
4
times the corresponding entry in the second vector.
The set is linearly dependent because the first vector is a multiple of the other vector. The entries in the first vector are
4
times the corresponding entry in the second vector.
The set is linearly dependent because neither vector is a multiple of the other vector. Two of the entries in the first vector are
4
times the corresponding entry in the second vector. But this multiple does not work for the third entries.
1.7.23
Determine whether the statement below is true or false. Justify the answer.
If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.
Choose the correct answer below.
The statement is false. If an indexed set of vectors, S, is linearly dependent, then it is only necessary that one of the vectors is a linear combination of the other vectors in the set.
The statement is false. If S is linearly dependent then there is at least one vector that is not a linear combination of the other vectors, but the others may be linear combinations of each other.
The statement is true. If an indexed set of vectors, S, is linearly dependent, then at least one of the vectors can be written as a linear combination of other vectors in the set. Using the basic properties of equality, each of the vectors in the linear combination can also be written as a linear combination of those vectors.
The statement is true. If S is linearly dependent then for each j,
vj,
a vector in S, is a linear combination of the preceding vectors in S.
1.7.23
Determine whether the statement below is true or false. Justify the answer.
If S is a linearly dependent set, then each vector is a linear combination of the other vectors in S.
Choose the correct answer below.
The statement is false. If S is linearly dependent then there is at least one vector that is not a linear combination of the other vectors, but the others may be linear combinations of each other.
The statement is true. If an indexed set of vectors, S, is linearly dependent, then at least one of the vectors can be written as a linear combination of other vectors in the set. Using the basic properties of equality, each of the vectors in the linear combination can also be written as a linear combination of those vectors.
The statement is false. If an indexed set of vectors, S, is linearly dependent, then it is only necessary that one of the vectors is a linear combination of the other vectors in the set.
The statement is true. If S is linearly dependent then for each j,
vj,
a vector in S, is a linear combination of the preceding vectors in S.
1.7.24
Determine whether the statement below is true or false. Justify the answer.
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
Choose the correct answer below.
The statement is false. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector.
The statement is false. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. If there are fewer vectors than entries in the vectors, then at least one of the vectors must be written as a linear combination of the others.
The statement is true. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly independent. One example is a set consisting of two vectors where one of the vectors is not a scalar multiple of the other vector.
The statement is true. If a set contains fewer vectors than there are entries in the vectors, then there are less variables than equations, so there cannot be any free variables in the equation
Ax=0.
1.7.24
Determine whether the statement below is true or false. Justify the answer.
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
Choose the correct answer below.
The statement is true. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly independent. One example is a set consisting of two vectors where one of the vectors is not a scalar multiple of the other vector.
The statement is false. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. If there are fewer vectors than entries in the vectors, then at least one of the vectors must be written as a linear combination of the others.
The statement is true. If a set contains fewer vectors than there are entries in the vectors, then there are less variables than equations, so there cannot be any free variables in the equation
Ax=0.
The statement is false. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector.
1.7.24
Determine whether the statement below is true or false. Justify the answer.
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
Choose the correct answer below.
The statement is true. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly independent. One example is a set consisting of two vectors where one of the vectors is not a scalar multiple of the other vector.
The statement is false. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector.
The statement is false. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. If there are fewer vectors than entries in the vectors, then at least one of the vectors must be written as a linear combination of the others.
The statement is true. If a set contains fewer vectors than there are entries in the vectors, then there are less variables than equations, so there cannot be any free variables in the equation
Ax=0.
1.7.24
Determine whether the statement below is true or false. Justify the answer.
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
Choose the correct answer below.
The statement is true. If a set contains fewer vectors than there are entries in the vectors, then there are less variables than equations, so there cannot be any free variables in the equation
Ax=0.
The statement is false. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector.
The statement is true. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly independent. One example is a set consisting of two vectors where one of the vectors is not a scalar multiple of the other vector.
The statement is false. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. If there are fewer vectors than entries in the vectors, then at least one of the vectors must be written as a linear combination of the others.
1.7.24
Determine whether the statement below is true or false. Justify the answer.
If a set contains fewer vectors than there are entries in the vectors, then the set is linearly independent.
Choose the correct answer below.
The statement is false. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly dependent. One example is a set consisting of two vectors where one of the vectors is a scalar multiple of the other vector.
The statement is true. There exists a set that contains fewer vectors than there are entries in the vectors that is linearly independent. One example is a set consisting of two vectors where one of the vectors is not a scalar multiple of the other vector.
The statement is false. A set of vectors is linearly independent if none of the vectors can be written as a linear combination of the others. If there are fewer vectors than entries in the vectors, then at least one of the vectors must be written as a linear combination of the others.
The statement is true. If a set contains fewer vectors than there are entries in the vectors, then there are less variables than equations, so there cannot be any free variables in the equation
Ax=0.
1.7.25
Determine whether the statement below is true or false. Justify the answer.
The columns of any
4×5
matrix are linearly dependent.
Choose the correct answer below.
The statement is false. If a matrix has more rows than columns then the columns of the matrix are linearly dependent.
The statement is true. When a
4×5
matrix is written in reduced echelon form, there will be at least one row of zeros, so the columns of the matrix are linearly dependent.
The statement is true. A
4×5
matrix has more columns than rows, and if a set contains more vectors than there are entries in each vector, then the set is linearly dependent.
The statement is false. If A is a
4×5
matrix then the matrix equation
Ax=0
is inconsistent because the reduced echelon augmented matrix has a row with all zeros except in the last column.
1.7.25
Determine whether the statement below is true or false. Justify the answer.
The columns of any
4×5
matrix are linearly dependent.
Choose the correct answer below.
The statement is true. When a
4×5
matrix is written in reduced echelon form, there will be at least one row of zeros, so the columns of the matrix are linearly dependent.
The statement is false. If a matrix has more rows than columns then the columns of the matrix are linearly dependent.
The statement is true. A
4×5
matrix has more columns than rows, and if a set contains more vectors than there are entries in each vector, then the set is linearly dependent.
The statement is false. If A is a
4×5
matrix then the matrix equation
Ax=0
is inconsistent because the reduced echelon augmented matrix has a row with all zeros except in the last column.
1.7.32
Describe the possible echelon forms of the following matrix.
A is a
3×3
matrix,
A=
a1a2a3
,
such that
{a1,
a2}
is linearly independent and
a3
is not in
Span{a1,
a2}.
Select all that apply. (Note that leading entries marked with an X may have any nonzero value and starred entries
(*)
may have any value including zero.)
X00
000
000
0X*
00X
000
00X
000
000
X**
0X*
00X
1.7.32
Describe the possible echelon forms of the following matrix.
A is a
4×3
matrix,
A=
a1a2a3
,
such that
{a1,
a2}
is linearly independent and
a3
is not in
Span{a1,
a2}.
Select all that apply. (Note that leading entries marked with an X may have any nonzero value and starred entries
(*)
may have any value including zero.)
0X*
00X
000
000
X**
X**
X**
0XX
X**
0X*
00X
000
X00
000
000
000
1.7.42
The following statement is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If the statement is true, give a justification.
If
v1,
v2,
v3
are in
3
and
v3
is not a linear combination of
v1,
v2,
then
{v1,
v2,
v3}
is linearly independent.
Fill in the blanks below.
The statement is
false.
Take
v1
and
v2
to be multiples of one vector and take
v3
to be not a multiple of that vector. For example,
v1=
1
1
1
,
v2=
2
2
2
,
v3=
1
0
0
.
Since at least one of the vectors is a linear combination of the other two, the three vectors are linearly
dependent.
1.7.42
The following statement is either true (in all cases) or false (for at least one example). If false, construct a specific example to show that the statement is not always true. Such an example is called a counterexample to the statement. If the statement is true, give a justification.
If
v1,
v2,
v3
are in
3
and
v3
is not a linear combination of
v1,
v2,
then
{v1,
v2,
v3}
is linearly independent.
Fill in the blanks below.
The statement is
false.
Take
v1
and
v2
to be multiples of one vector and take
v3
to be not a multiple of that vector. For example,
v1=
1
1
1
,
v2=
2
2
2
,
v3=
1
0
0
.
Since at least one of the vectors is a linear combination of the other two, the three vectors are linearly
dependent.
1.8.1
Let
A=
20
02
,
and define
T : 22
by
T(x)=Ax.
Find the images under T of
u=
2
6
and
v=
a
b
.
T(u)=
4
12
T(v)=
2a
2b
1.8.1
Let
A=
30
03
,
and define
T : 22
by
T(x)=Ax.
Find the images under T of
u=
4
5
and
v=
a
b
.
T(u)=
12
15
T(v)=
3a
3b
1.8.4
If T is defined by
T(x)=Ax,
find a vector x whose image under T is
b,
and determine whether x is unique. Let
A=
134
015
41316
and
b=
4
8
3
.
Find a single vector x whose image under T is
b.
x=
39
13
1
Is the vector x found in the previous step unique?
No, because there are no free variables in the system of equations.
Yes, because there is a free variable in the system of equations.
Yes, because there are no free variables in the system of equations.
No, because there is a free variable in the system of equations.
1.8.4
If T is defined by
T(x)=Ax,
find a vector x whose image under T is
b,
and determine whether x is unique. Let
A=
144
014
298
and
b=
4
1
3
.
Find a single vector x whose image under T is
b.
x=
20
5
1
Is the vector x found in the previous step unique?
Yes, because there are no free variables in the system of equations.
No, because there are no free variables in the system of equations.
Yes, because there is a free variable in the system of equations.
No, because there is a free variable in the system of equations.
1.8.4
If T is defined by
T(x)=Ax,
find a vector x whose image under T is
b,
and determine whether x is unique. Let
A=
123
013
379
and
b=
4
6
3
.
Find a single vector x whose image under T is
b.
x=
19
9
1
Is the vector x found in the previous step unique?
No, because there is a free variable in the system of equations.
Yes, because there are no free variables in the system of equations.
Yes, because there is a free variable in the system of equations.
No, because there are no free variables in the system of equations.
1.8.5
If T is defined by
T(x)=Ax,
find a vector x whose image under T is
b,
and determine whether x is unique. Let
A=
1719
42155
and
b=
3
5
.
Find a single vector x whose image under T is
b.
x=
4
1
0
Is the vector x found in the previous step unique?
Yes, because there are no free variables in the system of equations.
No, because there are no free variables in the system of equations.
Yes, because there is a free variable in the system of equations.
No, because there is a free variable in the system of equations.
1.8.5
If T is defined by
T(x)=Ax,
find a vector x whose image under T is
b,
and determine whether x is unique. Let
A=
1310
4616
and
b=
1
2
.
Find a single vector x whose image under T is
b.
x=
2
1
0
Is the vector x found in the previous step unique?
No, because there are no free variables in the system of equations.
No, because there is a free variable in the system of equations.
Yes, because there is a free variable in the system of equations.
Yes, because there are no free variables in the system of equations.
1.8.6
Find a vector x whose image under T, defined by
T(x)=Ax,
is
b,
and determine whether x is unique. Let
A=
136
41527
011
41325
,
b=
16
73
3
67
.
Find a single vector x whose image under T is
b.
x=
7
3
0
Is the vector x found in the previous step unique?
No, because there are no free variables in the system of equations.
Yes, because there are no free variables in the system of equations.
Yes, because there is a free variable in the system of equations.
No, because there is a free variable in the system of equations.
1.8.6
Find a vector x whose image under T, defined by
T(x)=Ax,
is
b,
and determine whether x is unique. Let
A=
137
2917
011
2715
,
b=
12
30
2
26
.
Find a single vector x whose image under T is
b.
x=
6
2
0
Is the vector x found in the previous step unique?
Yes, because there is a free variable in the system of equations.
Yes, because there are no free variables in the system of equations.
No, because there is a free variable in the system of equations.
No, because there are no free variables in the system of equations.
1.8.7
Let A be a
4×6
matrix. What must a and b be in order to define
T : ab
by
T(x)=Ax?
a=6
(Simplify your answer.)
b=4
(Simplify your answer.)
1.8.7
Let A be a
7×4
matrix. What must a and b be in order to define
T : ab
by
T(x)=Ax?
a=4
(Simplify your answer.)
b=7
(Simplify your answer.)
1.8.10
Find all x in
4
that are mapped into the zero vector by the transformation
xAx
for the given matrix A.
A=
12101
1024
0145
172624
Select the correct choice below and fill in the answer box within your choice.
There is only one vector, which is
x=.
x3
2
4
1
0
x1+x3+x4
x1+x2
1.8.10
Find all x in
4
that are mapped into the zero vector by the transformation
xAx
for the given matrix A.
A=
1261
1024
0123
24410
Select the correct choice below and fill in the answer box within your choice.
There is only one vector, which is
x=.
x1+x3+x4
x1+x2
x3
2
2
1
0
1.8.10
Find all x in
4
that are mapped into the zero vector by the transformation
xAx
for the given matrix A.
A=
1261
1024
0123
3148
Select the correct choice below and fill in the answer box within your choice.
There is only one vector, which is
x=.
x3
2
2
1
0
x1+x3+x4
x1+x2
1.8.10
Find all x in
4
that are mapped into the zero vector by the transformation
xAx
for the given matrix A.
A=
13133
1045
0134
161413
Select the correct choice below and fill in the answer box within your choice.
There is only one vector, which is
x=.
x3
4
3
1
0
x1+x3+x4
x1+x2
1.8.10
Find all x in
4
that are mapped into the zero vector by the transformation
xAx
for the given matrix A.
A=
142011
1044
0145
2145
Select the correct choice below and fill in the answer box within your choice.
There is only one vector, which is
x=.
x1+x3+x4
x3
4
4
1
0
x1+x2
1.8.12
Let
b=
13
1
3
19
,
and let A be the matrix
14125
1045
0123
342011
.
Is b in the range of the linear transformation
xAx?
Why or why not?
Is b in the range of the linear transformation? Why or why not?
No, b is not in the range of the linear transformation because the system represented by the appropriate augmented matrix is inconsistent.
Yes, b is in the range of the linear transformation because the system represented by the appropriate augmented matrix is consistent.
No, b is not in the range of the linear transformation because the system represented by the appropriate augmented matrix is consistent.
Yes, b is in the range of the linear transformation because the system represented by the appropriate augmented matrix is inconsistent.
1.8.12
Let
b=
3
2
2
3
,
and let A be the matrix
1270
1035
0123
41141
.
Is b in the range of the linear transformation
xAx?
Why or why not?
Is b in the range of the linear transformation? Why or why not?
Yes, b is in the range of the linear transformation because the system represented by the appropriate augmented matrix is consistent.
Yes, b is in the range of the linear transformation because the system represented by the appropriate augmented matrix is inconsistent.
No, b is not in the range of the linear transformation because the system represented by the appropriate augmented matrix is inconsistent.
No, b is not in the range of the linear transformation because the system represented by the appropriate augmented matrix is consistent.
1.8.12
Let
b=
13
1
3
19
,
and let A be the matrix
142015
1043
0145
342821
.
Is b in the range of the linear transformation
xAx?
Why or why not?
Is b in the range of the linear transformation? Why or why not?
Yes, b is in the range of the linear transformation because the system represented by the appropriate augmented matrix is consistent.
No, b is not in the range of the linear transformation because the system represented by the appropriate augmented matrix is inconsistent.
No, b is not in the range of the linear transformation because the system represented by the appropriate augmented matrix is consistent.
Yes, b is in the range of the linear transformation because the system represented by the appropriate augmented matrix is inconsistent.
1.8.17
Let T:
22
be a linear transformation that maps
u=
3
5
into
4
1
and maps
v=
3
3
into
1
3
.
Use the fact that T is linear to find the images under T of
3u,
4v,
and 3u+4v.
The image of
3u
is
12
3
.
The image of
4v
is
4
12
.
The image of
3u+4v
is
8
15
.
1.8.17
Let T:
22
be a linear transformation that maps
u=
3
4
into
6
1
and maps
v=
3
3
into
1
3
.
Use the fact that T is linear to find the images under T of
2u,
4v,
and 2u+4v.
The image of
2u
is
12
2
.
The image of
4v
is
4
12
.
The image of
2u+4v
is
8
14
.
1.8.20
Let
x=
x1
x2
,
v1=
8
2
,
and
v2=
9
6
,
and let T :
22
be a linear transformation that maps x into
x1v1+x2v2.
Find a matrix A such that
T(x)
is
Ax
for each
x.
A=
89
26
1.8.20
Let
x=
x1
x2
,
v1=
6
5
,
and
v2=
4
7
,
and let T :
22
be a linear transformation that maps x into
x1v1+x2v2.
Find a matrix A such that
T(x)
is
Ax
for each
x.
A=
64
57
1.8.21
Determine whether the statement below is true or false. Justify the answer.
A linear transformation is a special type of function.
Choose the correct answer below.
The statement is false. A linear transformation is not a function because it maps one vector x to more than one vector
T(x).
The statement is true. A linear transformation is a function from
to
that assigns to each vector x in
a vector
T(x)
in
.
The statement is false. A linear transformation is not a function because it maps more than one vector x to the same vector
T(x).
The statement is true. A linear transformation is a function from
n
to
m
that assigns to each vector x in
n
a vector
T(x)
in
m.
1.8.21
Determine whether the statement below is true or false. Justify the answer.
A linear transformation is a special type of function.
Choose the correct answer below.
The statement is false. A linear transformation is not a function because it maps more than one vector x to the same vector
T(x).
The statement is false. A linear transformation is not a function because it maps one vector x to more than one vector
T(x).
The statement is true. A linear transformation is a function from
to
that assigns to each vector x in
a vector
T(x)
in
.
The statement is true. A linear transformation is a function from
n
to
m
that assigns to each vector x in
n
a vector
T(x)
in
m.
1.8.22
Determine whether the statement below is true or false. Justify the answer.
Every matrix transformation is a linear transformation.
Choose the correct answer below.
The statement is false. Not every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is true. Every matrix transformation has the property
T(u+v)=T(u)+T(v),
but not all matrix transformations have the property
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is false. Every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is true. Every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
1.8.22
Determine whether the statement below is true or false. Justify the answer.
Every matrix transformation is a linear transformation.
Choose the correct answer below.
The statement is false. Every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is true. Every matrix transformation has the property
T(u+v)=T(u)+T(v),
but not all matrix transformations have the property
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is false. Not every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is true. Every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
1.8.22
Determine whether the statement below is true or false. Justify the answer.
Every matrix transformation is a linear transformation.
Choose the correct answer below.
The statement is false. Not every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is true. Every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is false. Every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is true. Every matrix transformation has the property
T(u+v)=T(u)+T(v),
but not all matrix transformations have the property
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
1.8.22
Determine whether the statement below is true or false. Justify the answer.
Every matrix transformation is a linear transformation.
Choose the correct answer below.
The statement is true. Every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is true. Every matrix transformation has the property
T(u+v)=T(u)+T(v),
but not all matrix transformations have the property
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is false. Every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
The statement is false. Not every matrix transformation has the properties
T(u+v)=T(u)+T(v)
and
T(cu)=cT(u)
for all u and
v,
in the domain of T and all scalars c.
1.8.23
Determine whether the statement below is true or false. Justify the answer.
If A is a
3×5
matrix and T is a transformation defined by
T(x)=Ax,
then the domain of T is
3.
Choose the correct answer below.
The statement is false. The domain is actually
,
because in the product
Ax,
if A is an
m×n
matrix then x must be a vector in
.
The statement is false. The domain is actually
5,
because in the product
Ax,
if A is an
m×n
matrix then x must be a vector in
n.
The statement is true. The domain is
3
because A has 3 columns, because in the product
Ax,
if A is an
m×n
matrix then x must be a vector in
m.
The statement is true. The domain is
3
because A has 3 rows, because in the product
Ax,
if A is an
m×n
matrix then x must be a vector in
m.
1.8.23
Determine whether the statement below is true or false. Justify the answer.
If A is a
3×5
matrix and T is a transformation defined by
T(x)=Ax,
then the domain of T is
3.
Choose the correct answer below.
The statement is false. The domain is actually
,
because in the product
Ax,
if A is an
m×n
matrix then x must be a vector in
.
The statement is true. The domain is
3
because A has 3 columns, because in the product
Ax,
if A is an
m×n
matrix then x must be a vector in
m.
The statement is false. The domain is actually
5,
because in the product
Ax,
if A is an
m×n
matrix then x must be a vector in
n.
The statement is true. The domain is
3
because A has 3 rows, because in the product
Ax,
if A is an
m×n
matrix then x must be a vector in
m.
1.8.25
Determine whether the statement below is true or false. Justify the answer.
If A is an
m×n
matrix, then the range of the transformation
xAx
is
m.
Choose the correct answer below.
The statement is false. The range of the transformation is
n
because the domain of the transformation is
m.
The statement is false. The range of the transformation is the set of all linear combinations of the columns of A, because each image of the transformation is of the form
Ax.
The statement is true. The range of the transformation is
m,
because each vector in
m
is a linear combination of the rows of A.
The statement is true. The range of the transformation is
m,
because each vector in
m
is a linear combination of the columns of A.
1.8.25
Determine whether the statement below is true or false. Justify the answer.
If A is an
m×n
matrix, then the range of the transformation
xAx
is
m.
Choose the correct answer below.
The statement is false. The range of the transformation is the set of all linear combinations of the columns of A, because each image of the transformation is of the form
Ax.
The statement is true. The range of the transformation is
m,
because each vector in
m
is a linear combination of the rows of A.
The statement is false. The range of the transformation is
n
because the domain of the transformation is
m.
The statement is true. The range of the transformation is
m,
because each vector in
m
is a linear combination of the columns of A.
1.8.28
Determine whether the statement below is true or false. Justify the answer.
A linear transformation preserves the operations of vector addition and scalar multiplication.
Choose the correct answer below.
The statement is true. The linear transformation
T(cu+dv)
is not the same as
cT(u)+dT(v)
in
m.
Therefore, vector addition and scalar multiplication are preserved.
The statement is false. The linear transformation
T(cu+dv)
is not the same as
cT(u)+dT(v)
in
m.
Therefore, vector addition and scalar multiplication are not preserved.
The statement is true. The linear transformation
T(cu+dv)
is the same as
cT(u)+dT(v)
in
m.
Therefore, vector addition and scalar multiplication are preserved.
The statement is false. The linear transformation
T(cu+dv)
is the same as
cT(u)+dT(v)
in
m.
Therefore, vector addition and scalar multiplication are not preserved.
1.8.28
Determine whether the statement below is true or false. Justify the answer.
A linear transformation preserves the operations of vector addition and scalar multiplication.
Choose the correct answer below.
The statement is false. The linear transformation
T(cu+dv)
is not the same as
cT(u)+dT(v)
in
m.
Therefore, vector addition and scalar multiplication are not preserved.
The statement is false. The linear transformation
T(cu+dv)
is the same as
cT(u)+dT(v)
in
m.
Therefore, vector addition and scalar multiplication are not preserved.
The statement is true. The linear transformation
T(cu+dv)
is the same as
cT(u)+dT(v)
in
m.
Therefore, vector addition and scalar multiplication are preserved.
The statement is true. The linear transformation
T(cu+dv)
is not the same as
cT(u)+dT(v)
in
m.
Therefore, vector addition and scalar multiplication are preserved.
1.8.29
Determine whether the statement below is true or false. Justify the answer.
A transformation T is linear if and only if
Tc1v1+c2v2=c1Tv1+c2Tv2
for all
v1
and
v2
in the domain of T and for all scalars
c1
and
c2.
Choose the correct answer below.
The statement is true. This equation correctly summarizes the properties necessary for a transformation to be linear.
The statement is false. A transformation T is linear if and only if
T(cu)=cT(u)
for all scalars c and all u in the domain of T.
The statement is false. A transformation T is linear if and only if
T(0)=0.
The statement is false. A transformation T is linear if and only if
T(u+v)=T(u)+T(v)
for all
u,
v in the domain of T.
1.8.29
Determine whether the statement below is true or false. Justify the answer.
A transformation T is linear if and only if
Tc1v1+c2v2=c1Tv1+c2Tv2
for all
v1
and
v2
in the domain of T and for all scalars
c1
and
c2.
Choose the correct answer below.
The statement is true. This equation correctly summarizes the properties necessary for a transformation to be linear.
The statement is false. A transformation T is linear if and only if
T(u+v)=T(u)+T(v)
for all
u,
v in the domain of T.
The statement is false. A transformation T is linear if and only if
T(0)=0.
The statement is false. A transformation T is linear if and only if
T(cu)=cT(u)
for all scalars c and all u in the domain of T.
1.8.29
Determine whether the statement below is true or false. Justify the answer.
A transformation T is linear if and only if
Tc1v1+c2v2=c1Tv1+c2Tv2
for all
v1
and
v2
in the domain of T and for all scalars
c1
and
c2.
Choose the correct answer below.
The statement is true. This equation correctly summarizes the properties necessary for a transformation to be linear.
The statement is false. A transformation T is linear if and only if
T(0)=0.
The statement is false. A transformation T is linear if and only if
T(cu)=cT(u)
for all scalars c and all u in the domain of T.
The statement is false. A transformation T is linear if and only if
T(u+v)=T(u)+T(v)
for all
u,
v in the domain of T.
1.8.31
Let
T : 22
be the linear transformation that reflects each point through the
x1-axis,
such that
A=
10
01
where
xAx.
Make two sketches that illustrate the two properties of a linear transformation.
Choose the correct graph below that shows the property
T(u+v)=T(u)+T(v).
v
u+v
u
T(u+v)
T(u)
T(v)
x1
x2
v
u+v
u
T(u)
T(v)
T(u+v)
x1
x2
v
u+v
u
T(v)
T(u)
T(u+v)
x1
x2
v
u+v
u
T(v)
T(u)
T(u+v)
x1
x2
Choose the correct graph below that shows the property
T(cu)=cT(u).
u
cu
T(cu)
T(u)
cT(u)
x1
x2
u
cu
T(cu)
T(u)
cT(u)
x1
x2
T(u)
T(cu)
u
cu
cT(u)
x1
x2
u
cu
T(u)
T(cu)
cT(u)
x1
x2
1.8.31
Let
T : 22
be the linear transformation that reflects each point through the
x1-axis,
such that
A=
10
01
where
xAx.
Make two sketches that illustrate the two properties of a linear transformation.
Choose the correct graph below that shows the property
T(u+v)=T(u)+T(v).
v
u+v
u
T(v)
T(u)
T(u+v)
x1
x2
v
u+v
u
T(u)
T(v)
T(u+v)
x1
x2
v
u+v
u
T(u+v)
T(u)
T(v)
x1
x2
v
u+v
u
T(v)
T(u)
T(u+v)
x1
x2
Choose the correct graph below that shows the property
T(cu)=cT(u).
u
cu
T(u)
T(cu)
cT(u)
x1
x2
u
cu
T(cu)
T(u)
cT(u)
x1
x2
u
cu
T(cu)
T(u)
cT(u)
x1
x2
T(u)
T(cu)
u
cu
cT(u)
x1
x2
1.8.39
Let T:
nm
be a linear transformation, and let
{v1,
v2,
v3}
be a linearly dependent set in
n.
Explain why the set
{T(v1),
T(v2),
T(v3)}
is linearly dependent.
Choose the correct answer below.
Given that the set
{v1,
v2,
v3}
is linearly dependent, there exist
c1,
c2,
c3,
not all zero, such that
c1v1+c2v2+c3v3=0.
It follows that
c1T(v1)+c2T(v2)+c3T(v3)0.
Therefore, the set
T(v1),
T(v2),
T(v3)}
is linearly dependent.
Given that the set
{v1,
v2,
v3}
is linearly dependent, there exist
c1,
c2,
c3,
not all zero, such that
c1v1+c2v2+c3v3=0.
It follows that
c1T(v1)+c2T(v2)+c3T(v3)=0.
Therefore, the set
T(v1),
T(v2),
T(v3)}
is linearly dependent.
Given that the set
{v1,
v2,
v3}
is linearly dependent, there exist
c1,
c2,
c3,
all zero, such that
c1v1+c2v2+c3v3=0.
It follows that
c1T(v1)+c2T(v2)+c3T(v3)=0.
Therefore, the set
T(v1),
T(v2),
T(v3)}
is linearly dependent.
Given that the set
{v1,
v2,
v3}
is linearly dependent, there exist
c1,
c2,
c3,
not all zero, such that
c1v1+c2v2+c3v30.
It follows that
c1T(v1)+c2T(v2)+c3T(v3)0.
Therefore, the set
T(v1),
T(v2),
T(v3)}
is linearly dependent.
1.8.39
Let T:
nm
be a linear transformation, and let
{v1,
v2,
v3}
be a linearly dependent set in
n.
Explain why the set
{T(v1),
T(v2),
T(v3)}
is linearly dependent.
Choose the correct answer below.
Given that the set
{v1,
v2,
v3}
is linearly dependent, there exist
c1,
c2,
c3,
not all zero, such that
c1v1+c2v2+c3v3=0.
It follows that
c1T(v1)+c2T(v2)+c3T(v3)0.
Therefore, the set
T(v1),
T(v2),
T(v3)}
is linearly dependent.
Given that the set
{v1,
v2,
v3}
is linearly dependent, there exist
c1,
c2,
c3,
not all zero, such that
c1v1+c2v2+c3v3=0.
It follows that
c1T(v1)+c2T(v2)+c3T(v3)=0.
Therefore, the set
T(v1),
T(v2),
T(v3)}
is linearly dependent.
Given that the set
{v1,
v2,
v3}
is linearly dependent, there exist
c1,
c2,
c3,
not all zero, such that
c1v1+c2v2+c3v30.
It follows that
c1T(v1)+c2T(v2)+c3T(v3)0.
Therefore, the set
T(v1),
T(v2),
T(v3)}
is linearly dependent.
Given that the set
{v1,
v2,
v3}
is linearly dependent, there exist
c1,
c2,
c3,
all zero, such that
c1v1+c2v2+c3v3=0.
It follows that
c1T(v1)+c2T(v2)+c3T(v3)=0.
Therefore, the set
T(v1),
T(v2),
T(v3)}
is linearly dependent.
1.9.1
Assume that T is a linear transformation. Find the standard matrix of T.
T:
24,
Te1=(2,
1,
2,
1),
and
Te2=(9,
5,
0, 0), where
e1=(1,0)
and
e2=(0,1).
A=
29
15
20
10
(Type an integer or decimal for each matrix element.)
1.9.1
Assume that T is a linear transformation. Find the standard matrix of T.
T:
24,
Te1=(2,
1,
2,
1),
and
Te2=(4,
7,
0, 0), where
e1=(1,0)
and
e2=(0,1).
A=
24
17
20
10
(Type an integer or decimal for each matrix element.)
1.9.1
Assume that T is a linear transformation. Find the standard matrix of T.
T:
24,
Te1=(4,
1,
4,
1),
and
Te2=(5,
3,
0, 0), where
e1=(1,0)
and
e2=(0,1).
A=
45
13
40
10
(Type an integer or decimal for each matrix element.)
1.9.1
Assume that T is a linear transformation. Find the standard matrix of T.
T:
24,
Te1=(8,
1,
8,
1),
and
Te2=(7,
5,
0, 0), where
e1=(1,0)
and
e2=(0,1).
A=
87
15
80
10
(Type an integer or decimal for each matrix element.)
1.9.15
Fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables.
???
???
???
x1
x2
x3
=
3x15x2
7x14x3
4x2+5x3
Fill in the missing entries of the matrix below.
350
704
045
x1
x2
x3
=
3x15x2
7x14x3
4x2+5x3
1.9.15
Fill in the missing entries of the matrix, assuming that the equation holds for all values of the variables.
???
???
???
x1
x2
x3
=
4x13x2
x14x3
7x2+8x3
Fill in the missing entries of the matrix below.
430
104
078
x1
x2
x3
=
4x13x2
x14x3
7x2+8x3
1.9.17
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that
x1,
x2,
... are not vectors but are entries in vectors.
Tx1,x2,x3,x4=x1+4x2, 0, 8x2+x4, x2x4
A=
1400
0000
0801
0101
(Type an integer or decimal for each matrix element.)
1.9.17
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that
x1,
x2,
... are not vectors but are entries in vectors.
Tx1,x2,x3,x4=x1+8x2, 0, 5x2+x4, x2x4
A=
1800
0000
0501
0101
(Type an integer or decimal for each matrix element.)
1.9.17
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that
x1,
x2,
... are not vectors but are entries in vectors.
Tx1,x2,x3,x4=x1+9x2, 0, 6x2+x4, x2x4
A=
1900
0000
0601
0101
(Type an integer or decimal for each matrix element.)
1.9.17
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that
x1,
x2,
... are not vectors but are entries in vectors.
Tx1,x2,x3,x4=x1+5x2, 0, 9x2+x4, x2x4
A=
1500
0000
0901
0101
(Type an integer or decimal for each matrix element.)
1.9.19
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that
x1,
x2,
... are not vectors but are entries in vectors.
Tx1,x2,x3=x12x2+7x3, x28x3
A=
127
018
(Type an integer or decimal for each matrix element.)
1.9.20
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that
x1,
x2,
... are not vectors but are entries in a vector.
Tx1,x2,x3,x4=4x1+x2x3+x4
                                                                (T:
4)
A=
4111
1.9.20
Show that T is a linear transformation by finding a matrix that implements the mapping. Note that
x1,
x2,
... are not vectors but are entries in a vector.
Tx1,x2,x3,x4=3x1+x22x34x4
                                                               (T:
4)
A=
3124
1.9.24
Determine whether the statement below is true or false. Justify the answer.
A mapping T:
nm
is one-to-one if each vector in
n
maps onto a unique vector in
m.
Choose the correct answer below.
The statement is true. A mapping T is said to be one-to-one if each x in
n
has at least one image for b in
m.
The statement is false. A mapping T is said to be one-to-one if each b in
m
is the image of at least one x in
n.
The statement is true. A mapping T is said to be one-to-one if each b in
m
is the image of exactly one x in
n.
The statement is false. A mapping T is said to be one-to-one if each b in
m
is the image of at most one x in
n.
1.9.24
Determine whether the statement below is true or false. Justify the answer.
A mapping T:
nm
is one-to-one if each vector in
n
maps onto a unique vector in
m.
Choose the correct answer below.
The statement is false. A mapping T is said to be one-to-one if each b in
m
is the image of at most one x in
n.
The statement is false. A mapping T is said to be one-to-one if each b in
m
is the image of at least one x in
n.
The statement is true. A mapping T is said to be one-to-one if each b in
m
is the image of exactly one x in
n.
The statement is true. A mapping T is said to be one-to-one if each x in
n
has at least one image for b in
m.
1.9.24
Determine whether the statement below is true or false. Justify the answer.
A mapping T:
nm
is one-to-one if each vector in
n
maps onto a unique vector in
m.
Choose the correct answer below.
The statement is false. A mapping T is said to be one-to-one if each b in
m
is the image of at least one x in
n.
The statement is false. A mapping T is said to be one-to-one if each b in
m
is the image of at most one x in
n.
The statement is true. A mapping T is said to be one-to-one if each b in
m
is the image of exactly one x in
n.
The statement is true. A mapping T is said to be one-to-one if each x in
n
has at least one image for b in
m.
1.9.24
Determine whether the statement below is true or false. Justify the answer.
A mapping T:
nm
is one-to-one if each vector in
n
maps onto a unique vector in
m.
Choose the correct answer below.
The statement is false. A mapping T is said to be one-to-one if each b in
m
is the image of at least one x in
n.
The statement is true. A mapping T is said to be one-to-one if each x in
n
has at least one image for b in
m.
The statement is true. A mapping T is said to be one-to-one if each b in
m
is the image of exactly one x in
n.
The statement is false. A mapping T is said to be one-to-one if each b in
m
is the image of at most one x in
n.
1.9.29
Determine whether the statement below is true or false. Justify the answer.
A mapping T:
nm
is onto
m
if every vector x in
n
maps onto some vector in
m.
Choose the correct answer below.
The statement is true. A transformation is onto when all possible values in the codomain, or range, are mapped to by some value in the domain.
The statement is false. A mapping T:
nm
is onto
m
if every vector in
m
is mapped onto by some vector x in
n.
The statement is false. A mapping T:
nm
is one-to-one if every vector x in
n
maps onto some vector in
m.
The statement is true. A transformation is onto when, for every b in the codomain, the matrix equation
Ax=b
has a unique solution.
1.9.29
Determine whether the statement below is true or false. Justify the answer.
A mapping T:
nm
is onto
m
if every vector x in
n
maps onto some vector in
m.
Choose the correct answer below.
The statement is true. A transformation is onto when all possible values in the codomain, or range, are mapped to by some value in the domain.
The statement is false. A mapping T:
nm
is one-to-one if every vector x in
n
maps onto some vector in
m.
The statement is false. A mapping T:
nm
is onto
m
if every vector in
m
is mapped onto by some vector x in
n.
The statement is true. A transformation is onto when, for every b in the codomain, the matrix equation
Ax=b
has a unique solution.
1.9.29
Determine whether the statement below is true or false. Justify the answer.
A mapping T:
nm
is onto
m
if every vector x in
n
maps onto some vector in
m.
Choose the correct answer below.
The statement is false. A mapping T:
nm
is onto
m
if every vector in
m
is mapped onto by some vector x in
n.
The statement is true. A transformation is onto when all possible values in the codomain, or range, are mapped to by some value in the domain.
The statement is false. A mapping T:
nm
is one-to-one if every vector x in
n
maps onto some vector in
m.
The statement is true. A transformation is onto when, for every b in the codomain, the matrix equation
Ax=b
has a unique solution.
1.9.29
Determine whether the statement below is true or false. Justify the answer.
A mapping T:
nm
is onto
m
if every vector x in
n
maps onto some vector in
m.
Choose the correct answer below.
The statement is true. A transformation is onto when all possible values in the codomain, or range, are mapped to by some value in the domain.
The statement is false. A mapping T:
nm
is onto
m
if every vector in
m
is mapped onto by some vector x in
n.
The statement is true. A transformation is onto when, for every b in the codomain, the matrix equation
Ax=b
has a unique solution.
The statement is false. A mapping T:
nm
is one-to-one if every vector x in
n
maps onto some vector in
m.
1.9.31
Determine whether the statement below is true or false. Justify the answer.
If A is a
3×2
matrix, then the transformation
xAx
cannot be one-to-one.
Choose the correct answer below.
The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If
Ax=b
does not have a free variable, then the transformation represented by A is one-to-one.
The statement is true. Transformations which have standard matrices which are not square cannot be one-to-one nor onto because they do not have pivot positions in every row and column.
The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. It does not matter what dimensions a vector is as long as it meets this requirement. The matrix A could be
1×4
and still represent a one-to-one transformation.
The statement is true. A transformation is one-to-one only if the columns of A are linearly independent and a
3×2
matrix cannot have linearly independent columns.
1.9.31
Determine whether the statement below is true or false. Justify the answer.
If A is a
3×2
matrix, then the transformation
xAx
cannot be one-to-one.
Choose the correct answer below.
The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If
Ax=b
does not have a free variable, then the transformation represented by A is one-to-one.
The statement is true. A transformation is one-to-one only if the columns of A are linearly independent and a
3×2
matrix cannot have linearly independent columns.
The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. It does not matter what dimensions a vector is as long as it meets this requirement. The matrix A could be
1×4
and still represent a one-to-one transformation.
The statement is true. Transformations which have standard matrices which are not square cannot be one-to-one nor onto because they do not have pivot positions in every row and column.
1.9.31
Determine whether the statement below is true or false. Justify the answer.
If A is a
3×2
matrix, then the transformation
xAx
cannot be one-to-one.
Choose the correct answer below.
The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. It does not matter what dimensions a vector is as long as it meets this requirement. The matrix A could be
1×4
and still represent a one-to-one transformation.
The statement is true. Transformations which have standard matrices which are not square cannot be one-to-one nor onto because they do not have pivot positions in every row and column.
The statement is true. A transformation is one-to-one only if the columns of A are linearly independent and a
3×2
matrix cannot have linearly independent columns.
The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If
Ax=b
does not have a free variable, then the transformation represented by A is one-to-one.
1.9.31
Determine whether the statement below is true or false. Justify the answer.
If A is a
3×2
matrix, then the transformation
xAx
cannot be one-to-one.
Choose the correct answer below.
The statement is true. A transformation is one-to-one only if the columns of A are linearly independent and a
3×2
matrix cannot have linearly independent columns.
The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. If
Ax=b
does not have a free variable, then the transformation represented by A is one-to-one.
The statement is false. A transformation is one-to-one if each vector in the codomain is mapped to by at most one vector in the domain. It does not matter what dimensions a vector is as long as it meets this requirement. The matrix A could be
1×4
and still represent a one-to-one transformation.
The statement is true. Transformations which have standard matrices which are not square cannot be one-to-one nor onto because they do not have pivot positions in every row and column.
1.9.33
Determine if the specified linear transformation is
(a)
one-to-one and
(b)
onto. Justify your answer.
Tx1,x2,x3,x4=(0,x3+x4,x1+x2,x1+x2)
a. Is the linear transformation one-to-one?
T is not one-to-one because the columns of the standard matrix A are linearly independent.
T is one-to-one because
T(x)=0
has only the trivial solution.
T is one-to-one because the column vectors are not scalar multiples of each other.
T is not one-to-one because the standard matrix A has a free variable.
b. Is the linear transformation onto?
T is onto because the standard matrix A does not have a pivot position for every row.
T is onto because the columns of the standard matrix A span
4.
T is not onto because the
first row
of the standard matrix A is all zeros.
T is not onto because the columns of the standard matrix A span
4.
1.9.33
Determine if the specified linear transformation is
(a)
one-to-one and
(b)
onto. Justify your answer.
Tx1,x2,x3,x4=(0,x3+x4,x1+x2,x2+x3)
a. Is the linear transformation one-to-one?
T is not one-to-one because the columns of the standard matrix A are linearly independent.
T is one-to-one because the column vectors are not scalar multiples of each other.
T is one-to-one because
T(x)=0
has only the trivial solution.
T is not one-to-one because the standard matrix A has a free variable.
b. Is the linear transformation onto?
T is onto because the standard matrix A does not have a pivot position for every row.
T is not onto because the columns of the standard matrix A span
4.
T is onto because the columns of the standard matrix A span
4.
T is not onto because the
first row
of the standard matrix A is all zeros.
1.9.35
Determine if the specified linear transformation is
(a)
one-to-one and
(b)
onto. Justify each answer.
Tx1,x2,x3=x15x2+6x3, x23x3
(a)
Is the linear transformation one-to-one?
T is not one-to-one because the columns of the standard matrix A are linearly dependent.
T is one-to-one because the column vectors are not scalar multiples of each other.
T is one-to-one because
T(x)=0
has only the trivial solution.
T is not one-to-one because the columns of the standard matrix A are linearly independent.
(b)
Is the linear transformation onto?
T is not onto because the columns of the standard matrix A span
2.
T is onto because the standard matrix A does not have a pivot position for every row.
T is onto because the columns of the standard matrix A span
2.
T is not onto because the standard matrix A does not have a pivot position for every row.
1.9.35
Determine if the specified linear transformation is
(a)
one-to-one and
(b)
onto. Justify each answer.
Tx1,x2,x3=x17x2+4x3, x29x3
(a)
Is the linear transformation one-to-one?
T is one-to-one because the column vectors are not scalar multiples of each other.
T is not one-to-one because the columns of the standard matrix A are linearly dependent.
T is not one-to-one because the columns of the standard matrix A are linearly independent.
T is one-to-one because
T(x)=0
has only the trivial solution.
(b)
Is the linear transformation onto?
T is onto because the columns of the standard matrix A span
2.
T is onto because the standard matrix A does not have a pivot position for every row.
T is not onto because the columns of the standard matrix A span
2.
T is not onto because the standard matrix A does not have a pivot position for every row.
1.9.45
Let T be the linear transformation whose standard matrix is given. Decide if T is a one-to-one mapping. Justify your answer.
7572
85425
610715
31914
Choose the correct answer below.
The transformation T is not one-to-one because the equation
T(x)=0
has only the trivial solution.
The transformation T is one-to-one because the equation
T(x)=0
has a nontrivial solution.
The transformation T is one-to-one because the equation
T(x)=0
has only the trivial solution.
The transformation T is not one-to-one because the equation
T(x)=0
has a nontrivial solution.
1.9.45
Let T be the linear transformation whose standard matrix is given. Decide if T is a one-to-one mapping. Justify your answer.
46412
92518
612426
51912
Choose the correct answer below.
The transformation T is not one-to-one because the equation
T(x)=0
has a nontrivial solution.
The transformation T is not one-to-one because the equation
T(x)=0
has only the trivial solution.
The transformation T is one-to-one because the equation
T(x)=0
has a nontrivial solution.
The transformation T is one-to-one because the equation
T(x)=0
has only the trivial solution.
1.9.45
Let T be the linear transformation whose standard matrix is given. Decide if T is a one-to-one mapping. Justify your answer.
5653
8456
78533
221124
Choose the correct answer below.
The transformation T is one-to-one because the equation
T(x)=0
has only the trivial solution.
The transformation T is not one-to-one because the equation
T(x)=0
has a nontrivial solution.
The transformation T is one-to-one because the equation
T(x)=0
has a nontrivial solution.
The transformation T is not one-to-one because the equation
T(x)=0
has only the trivial solution.
1.10.7
Write a matrix equation that determines the loop currents.
I1
I2
I3
I4
3 Ω
3 Ω
9 Ω
2 Ω
6 Ω
6 Ω
4 Ω
3 Ω
41 V25 V24 V11 V
For each matrix, let row 1 correspond to loop 1, row 2 correspond to loop 2, and so on. Also, enter positive values for positive voltages and negative values for negative voltages.
15903
91760
06156
30613
I1
I2
I3
I4
=
41
25
24
11
1.10.7
Write a matrix equation that determines the loop currents.
I1
I2
I3
I4
1 Ω
3 Ω
8 Ω
2 Ω
4 Ω
6 Ω
5 Ω
4 Ω
39 V28 V18 V12 V
For each matrix, let row 1 correspond to loop 1, row 2 correspond to loop 2, and so on. Also, enter positive values for positive voltages and negative values for negative voltages.
12803
81660
06144
30412
I1
I2
I3
I4
=
39
28
18
12
1.10.8
Write a matrix equation that determines the loop currents shown to the right.
I1
I2
I4
I3
I5
51483824431324235445
Complete the matrix equation.
122024
213404
041755
205113
445316
I1
I2
I3
I4
I5
=
51
38
24
48
0
1.10.8
Write a matrix equation that determines the loop currents shown to the right.
I1
I2
I4
I3
I5
51403529413324243443
Complete the matrix equation.
122024
211404
041333
203124
443415
I1
I2
I3
I4
I5
=
51
35
29
40
0
1.10.8
Write a matrix equation that determines the loop currents shown to the right.
I1
I2
I4
I3
I5
52453824222213223244
Complete the matrix equation.
81023
19402
041334
20392
324211
I1
I2
I3
I4
I5
=
52
38
24
45
0
1.10.8
Write a matrix equation that determines the loop currents shown to the right.
I1
I2
I4
I3
I5
55403328421213134335
Complete the matrix equation.
91013
19303
031445
10493
335314
I1
I2
I3
I4
I5
=
55
33
28
40
0
1.10.12
A car rental service in a certain town has a fleet of about
700
cars, at three locations. A car rented at one location may be returned to any of the three locations. The various fractions of cars returned to the three locations are shown in the matrix to the right. Suppose that on Monday there are
355
cars at the airport (or rented from there),
75
cars at the east side office, and
270
cars at the west side office. What will be the approximate distribution of cars on Wednesday?
Cars Rented From:
Airport
East
West
Returned To:
0.95      0.07      0.12
0.00      0.86      0.06
0.05      0.07      0.82
Airport
East
West
On Wednesday,
391
cars will be at the airport,
84
cars will be at the East location, and
225
cars will be at the West location.
(Round to the nearest integer as needed.)
1.10.12
A car rental service in a certain town has a fleet of about
600
cars, at three locations. A car rented at one location may be returned to any of the three locations. The various fractions of cars returned to the three locations are shown in the matrix to the right. Suppose that on Monday there are
195
cars at the airport (or rented from there),
55
cars at the east side office, and
350
cars at the west side office. What will be the approximate distribution of cars on Wednesday?
Cars Rented From:
Airport
East
West
Returned To:
0.95      0.04      0.10
0.00      0.92      0.05
0.05      0.04      0.85
Airport
East
West
On Wednesday,
245
cars will be at the airport,
78
cars will be at the East location, and
277
cars will be at the West location.
(Round to the nearest integer as needed.)
2.1.1
Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let
A=
201
464
,
B=
742
152
,
C=
23
22
,
and
D=
46
25
.
3A,
B3A,
AC, CD
Compute the matrix product
3A.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
3A=
603
121812
(Simplify your answer.)
The expression
3A
is undefined because A is not a square matrix.
The expression
3A
is undefined because matrices cannot be multiplied by numbers.
The expression
3A
is undefined because matrices cannot have negative coefficients.
Compute the martrix sum
B3A.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
B3A=
145
111314
(Simplify your answer.)
The expression
B3A
is undefined because B and A have different sizes.
The expression
B3A
is undefined because B and
3A
have different sizes.
The expression
B3A
is undefined because A is not a square matrix.
Compute the matrix product AC. Select the correct choice below and, if necessary, fill in the answer box within your choice.
AC=
(Simplify your answer.)
The expression AC is undefined because the number of rows in A is not equal to the number of rows in C.
The expression AC is undefined because the number of columns in A is not equal to the number of rows in C.
The expression AC is undefined because the number of rows in A is not equal to the number of columns in C.
Compute the matrix product CD. Select the correct choice below and, if necessary, fill in the answer box within your choice.
CD=
227
122
(Simplify your answer.)
The expression CD is undefined because the corresponding entries in C and D are not equal.
The expression CD is undefined because matrices with negative entries cannot be multiplied.
The expression CD is undefined because square matrices cannot be multiplied.
2.1.1
Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let
A=
301
363
,
B=
661
253
,
C=
13
11
,
and
D=
46
24
.
2A,
B2A,
AC, CD
Compute the matrix product
2A.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
2A=
602
6126
(Simplify your answer.)
The expression
2A
is undefined because matrices cannot have negative coefficients.
The expression
2A
is undefined because A is not a square matrix.
The expression
2A
is undefined because matrices cannot be multiplied by numbers.
Compute the martrix sum
B2A.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
B2A=
063
479
(Simplify your answer.)
The expression
B2A
is undefined because A is not a square matrix.
The expression
B2A
is undefined because B and A have different sizes.
The expression
B2A
is undefined because B and
2A
have different sizes.
Compute the matrix product AC. Select the correct choice below and, if necessary, fill in the answer box within your choice.
AC=
(Simplify your answer.)
The expression AC is undefined because the number of rows in A is not equal to the number of columns in C.
The expression AC is undefined because the number of columns in A is not equal to the number of rows in C.
The expression AC is undefined because the number of rows in A is not equal to the number of rows in C.
Compute the matrix product CD. Select the correct choice below and, if necessary, fill in the answer box within your choice.
CD=
218
62
(Simplify your answer.)
The expression CD is undefined because square matrices cannot be multiplied.
The expression CD is undefined because matrices with negative entries cannot be multiplied.
The expression CD is undefined because the corresponding entries in C and D are not equal.
2.1.1
Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let
A=
101
463
,
B=
842
242
,
C=
23
12
,
and
D=
45
14
.
3A,
B3A,
AC, CD
Compute the matrix product
3A.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
3A=
303
12189
(Simplify your answer.)
The expression
3A
is undefined because matrices cannot have negative coefficients.
The expression
3A
is undefined because A is not a square matrix.
The expression
3A
is undefined because matrices cannot be multiplied by numbers.
Compute the martrix sum
B3A.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
B3A=
545
101411
(Simplify your answer.)
The expression
B3A
is undefined because B and A have different sizes.
The expression
B3A
is undefined because B and
3A
have different sizes.
The expression
B3A
is undefined because A is not a square matrix.
Compute the matrix product AC. Select the correct choice below and, if necessary, fill in the answer box within your choice.
AC=
(Simplify your answer.)
The expression AC is undefined because the number of rows in A is not equal to the number of rows in C.
The expression AC is undefined because the number of columns in A is not equal to the number of rows in C.
The expression AC is undefined because the number of rows in A is not equal to the number of columns in C.
Compute the matrix product CD. Select the correct choice below and, if necessary, fill in the answer box within your choice.
CD=
522
63
(Simplify your answer.)
The expression CD is undefined because matrices with negative entries cannot be multiplied.
The expression CD is undefined because the corresponding entries in C and D are not equal.
The expression CD is undefined because square matrices cannot be multiplied.
2.1.2
Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let
A=
103
364
,
B=
742
233
,
C=
13
21
,
D=
34
25
,
and
E=
5
2
.
A+3B,
4C3E,
DB, EB
Compute the matrix sum
A+3B.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A+3B=
22123
9155
(Simplify your answer.)
The expression
A+3B
is undefined because B is not a square matrix.
The expression
A+3B
is undefined because A and
3B
have different sizes.
The expression
A+3B
is undefined because A is not a square matrix.
Compute the matrix sum
4C3E.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
4C3E=
(Simplify your answer.)
The expression
4C3E
is undefined because
4C
and
3E
have different sizes.
The expression
4C3E
is undefined because E is not a square matrix.
The expression
4C3E
is undefined because the number of rows in C is not equal to the number of columns in E.
Compute the matrix product DB. Select the correct choice below and, if necessary, fill in the answer box within your choice.
DB=
29246
4719
(Simplify your answer.)
The expression DB is undefined because the number of rows in D is not equal to the number of columns in B.
The expression DB is undefined because the number of columns in D is not equal to the number of columns in B.
The expression DB is undefined because the number of columns in D is not equal to the number of rows in B.
Compute the matrix product EB. Select the correct choice below and, if necessary, fill in the answer box within your choice.
EB=
(Simplify your answer.)
The expression EB is undefined because the number of columns in E is not equal to the number of rows in B.
The expression EB is undefined because the number of columns in E is not equal to the number of columns in B.
The expression EB is undefined because the number of rows in E is not equal to the number of columns in B.
2.1.2
Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let
A=
101
542
,
B=
841
243
,
C=
13
21
,
D=
46
24
,
and
E=
5
4
.
A+4B,
2C4E,
DB, EB
Compute the matrix sum
A+4B.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A+4B=
33163
132010
(Simplify your answer.)
The expression
A+4B
is undefined because B is not a square matrix.
The expression
A+4B
is undefined because A and
4B
have different sizes.
The expression
A+4B
is undefined because A is not a square matrix.
Compute the matrix sum
2C4E.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
2C4E=
(Simplify your answer.)
The expression
2C4E
is undefined because the number of rows in C is not equal to the number of columns in E.
The expression
2C4E
is undefined because E is not a square matrix.
The expression
2C4E
is undefined because
2C
and
4E
have different sizes.
Compute the matrix product DB. Select the correct choice below and, if necessary, fill in the answer box within your choice.
DB=
444014
8814
(Simplify your answer.)
The expression DB is undefined because the number of rows in D is not equal to the number of columns in B.
The expression DB is undefined because the number of columns in D is not equal to the number of columns in B.
The expression DB is undefined because the number of columns in D is not equal to the number of rows in B.
Compute the matrix product EB. Select the correct choice below and, if necessary, fill in the answer box within your choice.
EB=
(Simplify your answer.)
The expression EB is undefined because the number of rows in E is not equal to the number of columns in B.
The expression EB is undefined because the number of columns in E is not equal to the number of rows in B.
The expression EB is undefined because the number of columns in E is not equal to the number of columns in B.
2.1.2
Compute each matrix sum or product if it is defined. If an expression is undefined, explain why. Let
A=
201
453
,
B=
642
234
,
C=
23
12
,
D=
26
24
,
and
E=
5
4
.
A+3B,
3C2E,
DB, EB
Compute the matrix sum
A+3B.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
A+3B=
20125
10149
(Simplify your answer.)
The expression
A+3B
is undefined because B is not a square matrix.
The expression
A+3B
is undefined because A is not a square matrix.
The expression
A+3B
is undefined because A and
3B
have different sizes.
Compute the matrix sum
3C2E.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
3C2E=
(Simplify your answer.)
The expression
3C2E
is undefined because the number of rows in C is not equal to the number of columns in E.
The expression
3C2E
is undefined because E is not a square matrix.
The expression
3C2E
is undefined because
3C
and
2E
have different sizes.
Compute the matrix product DB. Select the correct choice below and, if necessary, fill in the answer box within your choice.
DB=
242620
4420
(Simplify your answer.)
The expression DB is undefined because the number of rows in D is not equal to the number of columns in B.
The expression DB is undefined because the number of columns in D is not equal to the number of columns in B.
The expression DB is undefined because the number of columns in D is not equal to the number of rows in B.
Compute the matrix product EB. Select the correct choice below and, if necessary, fill in the answer box within your choice.
EB=
(Simplify your answer.)
The expression EB is undefined because the number of columns in E is not equal to the number of columns in B.
The expression EB is undefined because the number of columns in E is not equal to the number of rows in B.
The expression EB is undefined because the number of rows in E is not equal to the number of columns in B.
2.1.4
Compute
A3I3
and
(3I3)A,
where
A=
623
346
313
.
A3I3=
323
316
310
(3I3)A=
1869
91218
939
2.1.4
Compute
A3I3
and
(3I3)A,
where
A=
612
435
211
.
A3I3=
312
405
212
(3I3)A=
1836
12915
633
2.1.4
Compute
A5I3
and
(5I3)A,
where
A=
622
527
422
.
A5I3=
122
537
423
(5I3)A=
301010
251035
201010
2.1.5
Compute the product AB by the definition of the product of matrices, where
Ab1
and
Ab2
are computed separately, and by the row-column rule for computing AB.
A=
23
14
54
,
B=
53
13
Set up the product
Ab1,
where
b1
is the first column of B.
Ab1=
23
14
54
5
1
(Use one answer box for A and use the other answer box for
b1.)
Calculate
Ab1,
where
b1
is the first column of B.
Ab1=
13
1
29
(Type an integer or decimal for each matrix element.)
Set up the product
Ab2,
where
b2
is the second column of B.
Ab2=
23
14
54
3
3
(Use one answer box for A and use the other answer box for
b2.)
Calculate
Ab2,
where
b2
is the second column of B.
Ab2=
15
9
27
(Type an integer or decimal for each matrix element.)
Determine the numerical expression for the first entry in the first column of AB using the row-column rule. Choose the correct answer below.
((2)(5))((3)(1)
2(5)3(1)
((2)+(5))((3)+(1)
2(5)+3(1)
Determine the product AB.
AB=
1315
19
2927
(Use integers or decimals for any numbers in the expression.)
2.1.5
Compute the product AB by the definition of the product of matrices, where
Ab1
and
Ab2
are computed separately, and by the row-column rule for computing AB.
A=
13
33
43
,
B=
51
12
Set up the product
Ab1,
where
b1
is the first column of B.
Ab1=
13
33
43
5
1
(Use one answer box for A and use the other answer box for
b1.)
Calculate
Ab1,
where
b1
is the first column of B.
Ab1=
8
12
23
(Type an integer or decimal for each matrix element.)
Set up the product
Ab2,
where
b2
is the second column of B.
Ab2=
13
33
43
1
2
(Use one answer box for A and use the other answer box for
b2.)
Calculate
Ab2,
where
b2
is the second column of B.
Ab2=
7
3
10
(Type an integer or decimal for each matrix element.)
Determine the numerical expression for the first entry in the first column of AB using the row-column rule. Choose the correct answer below.
1(5)3(1)
((1)+(5))((3)+(1)
((1)(5))((3)(1)
1(5)+3(1)
Determine the product AB.
AB=
87
123
2310
(Use integers or decimals for any numbers in the expression.)
2.1.5
Compute the product AB by the definition of the product of matrices, where
Ab1
and
Ab2
are computed separately, and by the row-column rule for computing AB.
A=
22
15
63
,
B=
43
33
Set up the product
Ab1,
where
b1
is the first column of B.
Ab1=
22
15
63
4
3
(Use one answer box for A and use the other answer box for
b1.)
Calculate
Ab1,
where
b1
is the first column of B.
Ab1=
14
11
33
(Type an integer or decimal for each matrix element.)
Set up the product
Ab2,
where
b2
is the second column of B.
Ab2=
22
15
63
3
3
(Use one answer box for A and use the other answer box for
b2.)
Calculate
Ab2,
where
b2
is the second column of B.
Ab2=
12
12
27
(Type an integer or decimal for each matrix element.)
Determine the numerical expression for the first entry in the first column of AB using the row-column rule. Choose the correct answer below.
2(4)2(3)
2(4)+2(3)
((2)(4))((2)(3)
((2)+(4))((2)+(3)
Determine the product AB.
AB=
1412
1112
3327
(Use integers or decimals for any numbers in the expression.)
2.1.7
If a matrix A is
5×2
and the product AB is
5×7,
what is the size of B?
The size of B is
2×7.
2.1.7
If a matrix A is
3×2
and the product AB is
3×7,
what is the size of B?
The size of B is
2×7.
2.1.8
How many rows does B have if BC is a
4×3
matrix?
Matrix B has
4
rows.
2.1.8
How many rows does B have if BC is a
9×7
matrix?
Matrix B has
9
rows.
2.1.9
Let
A=
23
11
and
B=
29
3k
.
What value(s) of k, if any, will make
AB=BA?
Select the correct choice below and, if necessary, fill in the answer box within your choice.
k=1
(Use a comma to separate answers as needed.)
No value of k will make
AB=BA
2.1.9
Let
A=
34
11
and
B=
28
2k
.
What value(s) of k, if any, will make
AB=BA?
Select the correct choice below and, if necessary, fill in the answer box within your choice.
k=2
(Use a comma to separate answers as needed.)
No value of k will make
AB=BA
2.1.9
Let
A=
32
12
and
B=
26
3k
.
What value(s) of k, if any, will make
AB=BA?
Select the correct choice below and, if necessary, fill in the answer box within your choice.
k=1
(Use a comma to separate answers as needed.)
No value of k will make
AB=BA
2.1.9
Let
A=
42
12
and
B=
24
2k
.
What value(s) of k, if any, will make
AB=BA?
Select the correct choice below and, if necessary, fill in the answer box within your choice.
k=2
(Use a comma to separate answers as needed.)
No value of k will make
AB=BA
2.1.10
Let
A=
46
812
,
B=
62
47
,
and
C=
31
25
.
Verify that
AB=AC
and yet
BC.
Show the calculations that are used to find the entries for matrix AB. Choose the correct answer below.
4(6)+(12)(7)4(2)+(12)(4)
8(6)+6(7)8(2)+6(4)
8(6)+6(4)8(2)+6(7)
4(6)+(12)(4)4(2)+(12)(7)
4(6)+6(4)4(2)+6(7)
8(6)+(12)(4)8(2)+(12)(7)
4(2)+6(4)4(6)+6(7)
8(2)+(12)(4)8(6)+(12)(7)
Show the calculations that are used to find the entries for matrix AC. Choose the correct answer below.
8(3)+6(2)8(1)+6(5)
4(3)+(12)(2)4(1)+(12)(5)
4(3)+6(2)4(1)+6(5)
8(3)+(12)(2)8(1)+(12)(5)
4(1)+6(2)4(3)+6(5)
8(1)+(12)(2)8(3)+(12)(5)
4(3)+(12)(5)4(1)+(12)(2)
8(3)+6(5)8(1)+6(2)
Verify that
AB=AC
by simplifying.
AB=AC=
034
068
(Type an integer or decimal for each matrix element.)
2.1.10
Let
A=
46
812
,
B=
84
69
,
and
C=
22
25
.
Verify that
AB=AC
and yet
BC.
Show the calculations that are used to find the entries for matrix AB. Choose the correct answer below.
4(4)+(6)(6)4(8)+(6)(9)
8(4)+12(6)8(8)+12(9)
8(8)+(6)(6)8(4)+(6)(9)
4(8)+12(6)4(4)+12(9)
4(8)+12(9)4(4)+12(6)
8(8)+(6)(9)8(4)+(6)(6)
4(8)+(6)(6)4(4)+(6)(9)
8(8)+12(6)8(4)+12(9)
Show the calculations that are used to find the entries for matrix AC. Choose the correct answer below.
4(2)+(6)(2)4(2)+(6)(5)
8(2)+12(2)8(2)+12(5)
4(2)+12(5)4(2)+12(2)
8(2)+(6)(5)8(2)+(6)(2)
8(2)+(6)(2)8(2)+(6)(5)
4(2)+12(2)4(2)+12(5)
4(2)+(6)(2)4(2)+(6)(5)
8(2)+12(2)8(2)+12(5)
Verify that
AB=AC
by simplifying.
AB=AC=
438
876
(Type an integer or decimal for each matrix element.)
2.1.10
Let
A=
46
812
,
B=
54
79
,
and
C=
12
35
.
Verify that
AB=AC
and yet
BC.
Show the calculations that are used to find the entries for matrix AB. Choose the correct answer below.
4(5)+6(7)4(4)+6(9)
8(5)+(12)(7)8(4)+(12)(9)
4(5)+(12)(9)4(4)+(12)(7)
8(5)+6(9)8(4)+6(7)
8(5)+6(7)8(4)+6(9)
4(5)+(12)(7)4(4)+(12)(9)
4(4)+6(7)4(5)+6(9)
8(4)+(12)(7)8(5)+(12)(9)
Show the calculations that are used to find the entries for matrix AC. Choose the correct answer below.
4(1)+6(3)4(2)+6(5)
8(1)+(12)(3)8(2)+(12)(5)
4(2)+6(3)4(1)+6(5)
8(2)+(12)(3)8(1)+(12)(5)
8(1)+6(3)8(2)+6(5)
4(1)+(12)(3)4(2)+(12)(5)
4(1)+(12)(5)4(2)+(12)(3)
8(1)+6(5)8(2)+6(3)
Verify that
AB=AC
by simplifying.
AB=AC=
2238
4476
(Type an integer or decimal for each matrix element.)
2.1.11
Let
A=
111
145
156
and
D=
600
050
002
.
Compute AD and DA. Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a
3×3
matrix B, not the identity matrix or zero matrix, such that
AB=BA.
Compute AD.
AD=
652
62010
62512
Compute DA.
DA=
666
52025
21012
Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Choose the correct answer below.
Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each column entry of A by the corresponding diagonal entry of D.
Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each column of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each row of A by the corresponding diagonal entry of D.
Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each row of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each column of A by the corresponding diagonal entry of D.
Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each row entry of A by the corresponding diagonal entry of D.
Find a
3×3
matrix B, not the identity matrix or zero matrix, such that
AB=BA.
Choose the correct answer below.
There is only one unique solution,
B=.
(Simplify your answers.)
There are infinitely many solutions. Any multiple of
I3
will satisfy the expression.
There does not exist a matrix, B, that will satisfy the expression.
2.1.11
Let
A=
111
157
176
and
D=
500
030
002
.
Compute AD and DA. Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a
3×3
matrix B, not the identity matrix or zero matrix, such that
AB=BA.
Compute AD.
AD=
532
51514
52112
Compute DA.
DA=
555
31521
21412
Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Choose the correct answer below.
Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each column of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each row of A by the corresponding diagonal entry of D.
Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each row entry of A by the corresponding diagonal entry of D.
Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each column entry of A by the corresponding diagonal entry of D.
Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each row of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each column of A by the corresponding diagonal entry of D.
Find a
3×3
matrix B, not the identity matrix or zero matrix, such that
AB=BA.
Choose the correct answer below.
There is only one unique solution,
B=.
(Simplify your answers.)
There are infinitely many solutions. Any multiple of
I3
will satisfy the expression.
There does not exist a matrix, B, that will satisfy the expression.
2.1.11
Let
A=
111
146
167
and
D=
700
040
003
.
Compute AD and DA. Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Find a
3×3
matrix B, not the identity matrix or zero matrix, such that
AB=BA.
Compute AD.
AD=
743
71618
72421
Compute DA.
DA=
777
41624
31821
Explain how the columns or rows of A change when A is multiplied by D on the right or on the left. Choose the correct answer below.
Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each row of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each column of A by the corresponding diagonal entry of D.
Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each row entry of A by the corresponding diagonal entry of D.
Right-multiplication (that is, multiplication on the right) by the diagonal matrix D multiplies each column of A by the corresponding diagonal entry of D. Left-multiplication by D multiplies each row of A by the corresponding diagonal entry of D.
Both right-multiplication (that is, multiplication on the right) and left-multiplication by the diagonal matrix D multiplies each column entry of A by the corresponding diagonal entry of D.
Find a
3×3
matrix B, not the identity matrix or zero matrix, such that
AB=BA.
Choose the correct answer below.
There is only one unique solution,
B=.
(Simplify your answers.)
There are infinitely many solutions. Any multiple of
I3
will satisfy the expression.
There does not exist a matrix, B, that will satisfy the expression.
2.1.12
Let
A=
36
510
.
Construct a
2×2
matrix B such that AB is the zero matrix. Use two different nonzero columns for B.
B=
24
12
2.1.12
Let
A=
515
39
.
Construct a
2×2
matrix B such that AB is the zero matrix. Use two different nonzero columns for B.
B=
63
21
2.1.16
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer.
If A and B are
3×3
matrices and
B=
b1b2b3
,
then
AB=
Ab1+Ab2+Ab3
.
Choose the correct answer below.
The statement is false. The matrix
Ab1+Ab2+Ab3
is the correct size matrix, but the plus signs should be minus signs.
The statement is false. The matrix
Ab1+Ab2+Ab3
is a
3×1
matrix, and AB must be a
3×3
matrix. The plus signs should be spaces between the 3 columns.
The statement is true. By the definition of matrix multiplication, if A is an
m×n
matrix and B is an
n×p
matrix, then the resulting matrix AB is the sum of the columns of A using the weights from the corresponding columns of B.
The statement is true. By the definition of matrix multiplication, if A is an
m×n
matrix and B is an
n×p
matrix, then
AB=A
b1b2...bp
=
Ab1+Ab2+...+Abp
.
2.1.16
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer.
If A and B are
3×3
matrices and
B=
b1b2b3
,
then
AB=
Ab1+Ab2+Ab3
.
Choose the correct answer below.
The statement is false. The matrix
Ab1+Ab2+Ab3
is the correct size matrix, but the plus signs should be minus signs.
The statement is true. By the definition of matrix multiplication, if A is an
m×n
matrix and B is an
n×p
matrix, then
AB=A
b1b2...bp
=
Ab1+Ab2+...+Abp
.
The statement is true. By the definition of matrix multiplication, if A is an
m×n
matrix and B is an
n×p
matrix, then the resulting matrix AB is the sum of the columns of A using the weights from the corresponding columns of B.
The statement is false. The matrix
Ab1+Ab2+Ab3
is a
3×1
matrix, and AB must be a
3×3
matrix. The plus signs should be spaces between the 3 columns.
2.1.19
Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Determine whether the statement below is true or false. Justify the answer.
AB+AC=A(B+C)
Choose the correct answer below.
The statement is false. The distributive law does not apply to matrix multiplication.
The statement is true. The multiplication laws for matrices are the same as those for real numbers.
The statement is false. The distributive law for matrices states that
A(B+C)=BA+CA.
The statement is true. The distributive law for matrices states that
A(B+C)=AB+AC.
2.1.19
Let A, B, and C be arbitrary matrices for which the indicated sums and products are defined. Determine whether the statement below is true or false. Justify the answer.
AB+AC=A(B+C)
Choose the correct answer below.
The statement is true. The multiplication laws for matrices are the same as those for real numbers.
The statement is true. The distributive law for matrices states that
A(B+C)=AB+AC.
The statement is false. The distributive law does not apply to matrix multiplication.
The statement is false. The distributive law for matrices states that
A(B+C)=BA+CA.
2.1.20
Let A and B be arbitrary matrices for which the indicated sum is defined. Determine whether the statement below is true or false. Justify the answer.
AT+BT=(A+B)T
Choose the correct answer below.
The statement is false. The transpose property states that
(A+B)T=ATBT.
The statement is true. The transpose property for matrices is the same as for algebraic exponents of real numbers.
The statement is true. The transpose property states that
(A+B)T=AT+BT.
The statement is false. The transpose property is inapplicable here.
2.1.20
Let A and B be arbitrary matrices for which the indicated sum is defined. Determine whether the statement below is true or false. Justify the answer.
AT+BT=(A+B)T
Choose the correct answer below.
The statement is true. The transpose property for matrices is the same as for algebraic exponents of real numbers.
The statement is false. The transpose property states that
(A+B)T=ATBT.
The statement is false. The transpose property is inapplicable here.
The statement is true. The transpose property states that
(A+B)T=AT+BT.
2.1.20
Let A and B be arbitrary matrices for which the indicated sum is defined. Determine whether the statement below is true or false. Justify the answer.
AT+BT=(A+B)T
Choose the correct answer below.
The statement is false. The transpose property is inapplicable here.
The statement is true. The transpose property states that
(A+B)T=AT+BT.
The statement is false. The transpose property states that
(A+B)T=ATBT.
The statement is true. The transpose property for matrices is the same as for algebraic exponents of real numbers.
2.1.21
Let A, B, and C be arbitrary matrices for which the indicated products are defined. Determine whether the statement below is true or false. Justify the answer.
(AB)C=(AC)B
Choose the correct answer below.
The statement is false. The associative law of multiplication for matrices states that
(AB)C=B(AC).
The statement is false. The associative law of multiplication for matrices states that
A(BC)=(AB)C.
The statement is true. The associative law of multiplication for matrices states that
(AB)C=(AC)B.
The statement is true. The associative law of multiplication for matrices states that
(AB)C=B(AC).
2.1.21
Let A, B, and C be arbitrary matrices for which the indicated products are defined. Determine whether the statement below is true or false. Justify the answer.
(AB)C=(AC)B
Choose the correct answer below.
The statement is false. The associative law of multiplication for matrices states that
A(BC)=(AB)C.
The statement is false. The associative law of multiplication for matrices states that
(AB)C=B(AC).
The statement is true. The associative law of multiplication for matrices states that
(AB)C=(AC)B.
The statement is true. The associative law of multiplication for matrices states that
(AB)C=B(AC).
2.1.21
Let A, B, and C be arbitrary matrices for which the indicated products are defined. Determine whether the statement below is true or false. Justify the answer.
(AB)C=(AC)B
Choose the correct answer below.
The statement is true. The associative law of multiplication for matrices states that
(AB)C=(AC)B.
The statement is false. The associative law of multiplication for matrices states that
(AB)C=B(AC).
The statement is true. The associative law of multiplication for matrices states that
(AB)C=B(AC).
The statement is false. The associative law of multiplication for matrices states that
A(BC)=(AB)C.
2.1.22
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer.
(AB)T=ATBT
Choose the correct answer below.
The statement is false. The transpose of the product of two matrices is the product of the transposes of the individual matrices in reverse order, or
(AB)T=BTAT.
The statement is true. The transpose of the product of two matrices is the product of the transposes of the individual matrices in the same order, or
(AB)T=ATBT.
The statement is true. Matrix multiplication is not commutative so the products must remain in the same order.
The statement is false. The transpose of the product of two matrices is the product of the transpose of the first matrix and the second matrix, or
(AB)T=ATB.
2.1.22
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer.
(AB)T=ATBT
Choose the correct answer below.
The statement is true. Matrix multiplication is not commutative so the products must remain in the same order.
The statement is false. The transpose of the product of two matrices is the product of the transposes of the individual matrices in reverse order, or
(AB)T=BTAT.
The statement is true. The transpose of the product of two matrices is the product of the transposes of the individual matrices in the same order, or
(AB)T=ATBT.
The statement is false. The transpose of the product of two matrices is the product of the transpose of the first matrix and the second matrix, or
(AB)T=ATB.
2.1.22
Let A and B be arbitrary matrices for which the indicated product is defined. Determine whether the statement below is true or false. Justify the answer.
(AB)T=ATBT
Choose the correct answer below.
The statement is false. The transpose of the product of two matrices is the product of the transposes of the individual matrices in reverse order, or
(AB)T=BTAT.
The statement is true. The transpose of the product of two matrices is the product of the transposes of the individual matrices in the same order, or
(AB)T=ATBT.
The statement is true. Matrix multiplication is not commutative so the products must remain in the same order.
The statement is false. The transpose of the product of two matrices is the product of the transpose of the first matrix and the second matrix, or
(AB)T=ATB.
2.1.26
Suppose the first two columns,
b1
and
b2,
of B are equal. What can you say about the columns of AB (if AB is defined)? Why?
Choose the correct answer below.
The first two columns of AB will be equal only if the first two rows of A are equal.
The first two columns of AB are
Ab1
and
Ab2.
They are equal since
b1
and
b2
are equal.
The first two columns of AB will be equal only if the first two columns of A are equal.
Nothing can be determined about the columns of AB since the entries of A are unknown.
2.1.26
Suppose the first two columns,
b1
and
b2,
of B are equal. What can you say about the columns of AB (if AB is defined)? Why?
Choose the correct answer below.
The first two columns of AB are
Ab1
and
Ab2.
They are equal since
b1
and
b2
are equal.
The first two columns of AB will be equal only if the first two columns of A are equal.
Nothing can be determined about the columns of AB since the entries of A are unknown.
The first two columns of AB will be equal only if the first two rows of A are equal.
2.1.47
Let
S=
09000
00900
00090
00009
00000
.
Compute
Sk
for
k=2, ..., 6.
S2=
008100
000810
000081
00000
00000
S3=
0007290
0000729
00000
00000
00000
S4=
00006561
00000
00000
00000
00000
S5=
00000
00000
00000
00000
00000
S6=
00000
00000
00000
00000
00000
2.1.47
Let
S=
01000
00100
00010
00001
00000
.
Compute
Sk
for
k=2, ..., 6.
S2=
00100
00010
00001
00000
00000
S3=
00010
00001
00000
00000
00000
S4=
00001
00000
00000
00000
00000
S5=
00000
00000
00000
00000
00000
S6=
00000
00000
00000
00000
00000
2.2.1
Find the inverse of the matrix.
75
84
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
13512
23712
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
2.2.1
Find the inverse of the matrix.
39
72
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
257319
757119
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
2.2.1
Find the inverse of the matrix.
48
92
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
13218
964116
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
2.2.8
Use the given inverse of the coefficient matrix to solve the following system.
7x1+2x2
=
6
A1=
11
372
6x12x2
=
2
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
x1=4
and
x2=11
(Simplify your answers.)
There is no solution.
2.2.8
Use the given inverse of the coefficient matrix to solve the following system.
7x1+2x2
=
8
A1=
11
372
6x12x2
=
4
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
x1=4
and
x2=10
(Simplify your answers.)
There is no solution.
2.2.9
Let
A=
12
818
,
b1=
1
16
,
b2=
5
34
,
b3=
2
12
,
and
b4=
6
40
.
a. Find
A1
and use it solve the four equations
Ax=b1,
Ax=b2
,Ax=b3,
and
Ax=b4.
b. The four equations in part (a) can be solved by the same set of operations, since the coefficient matrix is the same in each case. Solve the four equations in part (a) by row reducing the augmented matrix [A
b1
b2
b3
b4].
a. Find
A1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice
The inverse matrix is
A1=
91
412
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
Solve
Ax=b1.
x=
7
4
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b2.
x=
11
3
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b3.
x=
6
2
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b4.
x=
14
4
(Type an integer or simplified fraction for each matrix element.)
b. Solve the four equations by row reducing the augmented matrix [A
b1
b2
b3
b4].
Write the augmented matrix [A
b1
b2
b3
b4]
in reduced echelon form.
10711614
014324
(Type an integer or simplified fraction for each matrix element.)
2.2.9
Let
A=
12
818
,
b1=
1
2
,
b2=
5
34
,
b3=
3
22
,
and
b4=
4
22
.
a. Find
A1
and use it solve the four equations
Ax=b1,
Ax=b2
,Ax=b3,
and
Ax=b4.
b. The four equations in part (a) can be solved by the same set of operations, since the coefficient matrix is the same in each case. Solve the four equations in part (a) by row reducing the augmented matrix [A
b1
b2
b3
b4].
a. Find
A1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice
The inverse matrix is
A1=
91
412
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
Solve
Ax=b1.
x=
7
3
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b2.
x=
11
3
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b3.
x=
5
1
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b4.
x=
14
5
(Type an integer or simplified fraction for each matrix element.)
b. Solve the four equations by row reducing the augmented matrix [A
b1
b2
b3
b4].
Write the augmented matrix [A
b1
b2
b3
b4]
in reduced echelon form.
10711514
013315
(Type an integer or simplified fraction for each matrix element.)
2.2.9
Let
A=
12
716
,
b1=
2
8
,
b2=
1
3
,
b3=
2
20
,
and
b4=
5
27
.
a. Find
A1
and use it solve the four equations
Ax=b1,
Ax=b2
,Ax=b3,
and
Ax=b4.
b. The four equations in part (a) can be solved by the same set of operations, since the coefficient matrix is the same in each case. Solve the four equations in part (a) by row reducing the augmented matrix [A
b1
b2
b3
b4].
a. Find
A1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice
The inverse matrix is
A1=
81
7212
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
Solve
Ax=b1.
x=
8
3
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b2.
x=
11
5
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b3.
x=
4
3
(Type an integer or simplified fraction for each matrix element.)
Solve
Ax=b4.
x=
13
4
(Type an integer or simplified fraction for each matrix element.)
b. Solve the four equations by row reducing the augmented matrix [A
b1
b2
b3
b4].
Write the augmented matrix [A
b1
b2
b3
b4]
in reduced echelon form.
10811413
013534
(Type an integer or simplified fraction for each matrix element.)
2.2.10
Use matrix algebra to show that if A is invertible and D satisfies
AD=I,
then
D=A1.
Choose the correct answer below.
Add
A1
to both sides of the equation
AD=I
to obtain
A1+AD=A1+I,
ID=A1,
and
D=A1.
Right-multiply each side of the equation
AD=I
by
A1
to obtain
ADA1=IA1,
DI=A1,
and
D=A1.
Left-multiply each side of the equation
AD=I
by
A1
to obtain
A1AD=A1I,
ID=A1,
and
D=A1.
Add
A1
to both sides of the equation
AD=I
to obtain
AD+A1=I+A1,
DI=A1,
and
D=A1.
2.2.10
Use matrix algebra to show that if A is invertible and D satisfies
AD=I,
then
D=A1.
Choose the correct answer below.
Right-multiply each side of the equation
AD=I
by
A1
to obtain
ADA1=IA1,
DI=A1,
and
D=A1.
Left-multiply each side of the equation
AD=I
by
A1
to obtain
A1AD=A1I,
ID=A1,
and
D=A1.
Add
A1
to both sides of the equation
AD=I
to obtain
A1+AD=A1+I,
ID=A1,
and
D=A1.
Add
A1
to both sides of the equation
AD=I
to obtain
AD+A1=I+A1,
DI=A1,
and
D=A1.
2.2.11
Determine whether the statement below is true or false. Justify the answer.
In order for a matrix B to be the inverse of A, both equations
AB=I
and
BA=I
must be true.
Choose the correct answer below.
The statement is true. The product of a matrix and its inverse is the identity matrix.
The statement is true. Since
AB=BA,
AB=I
if and only if
BA=I.
The statement is false. If
AB=I
and
BC=I,
then A is one inverse of B and C is possibly another inverse of B.
The statement is false. It's possible that the product AB is defined and equals
I,
yet the product BA is not defined.
2.2.11
Determine whether the statement below is true or false. Justify the answer.
In order for a matrix B to be the inverse of A, both equations
AB=I
and
BA=I
must be true.
Choose the correct answer below.
The statement is false. If
AB=I
and
BC=I,
then A is one inverse of B and C is possibly another inverse of B.
The statement is false. It's possible that the product AB is defined and equals
I,
yet the product BA is not defined.
The statement is true. The product of a matrix and its inverse is the identity matrix.
The statement is true. Since
AB=BA,
AB=I
if and only if
BA=I.
2.2.12
Determine whether the statement below is true or false. Justify the answer.
A product of invertible
n×n
matrices is invertible, and the inverse of the product is the product of their inverses in the same order.
Choose the correct answer below.
The statement is true. If A and B are invertible matrices, then
(AB)1=A1B1.
The statement is false. If A and B are invertible matrices, then
(AB)1=B1A1.
The statement is true. Since invertible matrices commute,
(AB)1=B1A1=A1B1.
The statement is false. If A and B are invertible matrices, then
(AB)1=BA1B1.
2.2.13
Determine whether the statement below is true or false. Justify the answer.
If A and B are
n×n
and invertible, then
A1B1
is the inverse of AB.
Choose the correct answer below.
The statement is false. The inverse of AB is
1AB.
The statement is false. The inverse of AB is
B1A1.
The statement is true. The following equalities hold,
(AB)1=(BA)1=A1B1.
The statement is true. The following equalities hold,
A1B1=B1A1=(AB)1.
2.2.13
Determine whether the statement below is true or false. Justify the answer.
If A and B are
n×n
and invertible, then
A1B1
is the inverse of AB.
Choose the correct answer below.
The statement is false. The inverse of AB is
B1A1.
The statement is true. The following equalities hold,
(AB)1=(BA)1=A1B1.
The statement is true. The following equalities hold,
A1B1=B1A1=(AB)1.
The statement is false. The inverse of AB is
1AB.
2.2.14
Determine whether the statement below is true or false. Justify the answer.
If A is invertible, then the inverse of
A1
is A itself.
Choose the correct answer below.
The statement is true. A is invertible if and only if
A=A1.
Since A is invertible,
A=A1.
Since
A=A1,
A1
is invertible, and A is the inverse of
A1.
The statement is false. It does not follow from the fact that A is invertible that
A1
is also invertible.
The statement is false. Since inverses are not unique, it is possible that
BA
is the inverse of
A1.
The statement is true. Since
A1
is the inverse of A,
A1A=I=AA1.
Since
A1A=I=AA1,
A is the inverse of
A1.
2.2.14
Determine whether the statement below is true or false. Justify the answer.
If A is invertible, then the inverse of
A1
is A itself.
Choose the correct answer below.
The statement is true. Since
A1
is the inverse of A,
A1A=I=AA1.
Since
A1A=I=AA1,
A is the inverse of
A1.
The statement is false. It does not follow from the fact that A is invertible that
A1
is also invertible.
The statement is true. A is invertible if and only if
A=A1.
Since A is invertible,
A=A1.
Since
A=A1,
A1
is invertible, and A is the inverse of
A1.
The statement is false. Since inverses are not unique, it is possible that
BA
is the inverse of
A1.
2.2.14
Determine whether the statement below is true or false. Justify the answer.
If A is invertible, then the inverse of
A1
is A itself.
Choose the correct answer below.
The statement is true. Since
A1
is the inverse of A,
A1A=I=AA1.
Since
A1A=I=AA1,
A is the inverse of
A1.
The statement is false. It does not follow from the fact that A is invertible that
A1
is also invertible.
The statement is false. Since inverses are not unique, it is possible that
BA
is the inverse of
A1.
The statement is true. A is invertible if and only if
A=A1.
Since A is invertible,
A=A1.
Since
A=A1,
A1
is invertible, and A is the inverse of
A1.
2.2.14
Determine whether the statement below is true or false. Justify the answer.
If A is invertible, then the inverse of
A1
is A itself.
Choose the correct answer below.
The statement is false. Since inverses are not unique, it is possible that
BA
is the inverse of
A1.
The statement is true. A is invertible if and only if
A=A1.
Since A is invertible,
A=A1.
Since
A=A1,
A1
is invertible, and A is the inverse of
A1.
The statement is false. It does not follow from the fact that A is invertible that
A1
is also invertible.
The statement is true. Since
A1
is the inverse of A,
A1A=I=AA1.
Since
A1A=I=AA1,
A is the inverse of
A1.
2.2.14
Determine whether the statement below is true or false. Justify the answer.
If A is invertible, then the inverse of
A1
is A itself.
Choose the correct answer below.
The statement is false. Since inverses are not unique, it is possible that
BA
is the inverse of
A1.
The statement is true. A is invertible if and only if
A=A1.
Since A is invertible,
A=A1.
Since
A=A1,
A1
is invertible, and A is the inverse of
A1.
The statement is false. It does not follow from the fact that A is invertible that
A1
is also invertible.
The statement is true. Since
A1
is the inverse of A,
A1A=I=AA1.
Since
A1A=I=AA1,
A is the inverse of
A1.
2.2.15
Determine whether the statement below is true or false. Justify the answer.
If
A=
ab
cd
and
abcd0,
then A is invertible.
Choose the correct answer below.
The statement is false. If A is invertible, then
ab=cd.
The statement is true. The matrix
A=
ab
cd
is invertible if and only if
ac
and
bd.
The statement is true. The inverse of A is
A1=1abcd
db
ca
.
This expression is always defined when
abcd0.
The statement is false. If
adbc0,
then A is invertible.
2.2.15
Determine whether the statement below is true or false. Justify the answer.
If
A=
ab
cd
and
abcd0,
then A is invertible.
Choose the correct answer below.
The statement is true. The inverse of A is
A1=1abcd
db
ca
.
This expression is always defined when
abcd0.
The statement is false. If
adbc0,
then A is invertible.
The statement is false. If A is invertible, then
ab=cd.
The statement is true. The matrix
A=
ab
cd
is invertible if and only if
ac
and
bd.
2.2.16
Determine whether the statement below is true or false. Justify the answer.
If
A=
ab
cd
and
ad=bc,
then A is not invertible.
Choose the correct answer below.
The statement is false. If A is invertible, then
ad=bc.
The statement is false. If
ad=bc,
then A is invertible.
The statement is true. If
ad=bc
then
adbc=0,
and
1adbc
db
ca
is undefined.
The statement is true. The matrix
A=
ab
cd
is invertible if and only if
ab
and
bd.
2.2.16
Determine whether the statement below is true or false. Justify the answer.
If
A=
ab
cd
and
ad=bc,
then A is not invertible.
Choose the correct answer below.
The statement is true. The matrix
A=
ab
cd
is invertible if and only if
ab
and
bd.
The statement is true. If
ad=bc
then
adbc=0,
and
1adbc
db
ca
is undefined.
The statement is false. If
ad=bc,
then A is invertible.
The statement is false. If A is invertible, then
ad=bc.
2.2.17
Determine whether the statement below is true or false. Justify the answer.
If A is an invertible
n×n
matrix, then the equation
Ax=b
is consistent for each
b
in
n.
Choose the correct answer below.
The statement is false. The matrix A satisfies
Ax=b
if and only if A is row equivalent to the identity matrix, and not every matrix that is row equivalent to the identity matrix is invertible.
The statement is true. Since A is invertible,
A1b
exists for all
b
in
n.
Define
x=A1b.
Then
Ax=b.
The statement is true. Since A is invertible,
A1b=x
for all
x
in
n.
Multiply both sides by A and the result is
Ax=b.
The statement is false. The matrix A is invertible if and only if A is row equivalent to the identity matrix, and not every matrix A satisfying
Ax=b
is row equivalent to the identity matrix.
2.2.17
Determine whether the statement below is true or false. Justify the answer.
If A is an invertible
n×n
matrix, then the equation
Ax=b
is consistent for each
b
in
n.
Choose the correct answer below.
The statement is false. The matrix A is invertible if and only if A is row equivalent to the identity matrix, and not every matrix A satisfying
Ax=b
is row equivalent to the identity matrix.
The statement is true. Since A is invertible,
A1b=x
for all
x
in
n.
Multiply both sides by A and the result is
Ax=b.
The statement is false. The matrix A satisfies
Ax=b
if and only if A is row equivalent to the identity matrix, and not every matrix that is row equivalent to the identity matrix is invertible.
The statement is true. Since A is invertible,
A1b
exists for all
b
in
n.
Define
x=A1b.
Then
Ax=b.
2.2.17
Determine whether the statement below is true or false. Justify the answer.
If A is an invertible
n×n
matrix, then the equation
Ax=b
is consistent for each
b
in
n.
Choose the correct answer below.
The statement is true. Since A is invertible,
A1b=x
for all
x
in
n.
Multiply both sides by A and the result is
Ax=b.
The statement is false. The matrix A satisfies
Ax=b
if and only if A is row equivalent to the identity matrix, and not every matrix that is row equivalent to the identity matrix is invertible.
The statement is true. Since A is invertible,
A1b
exists for all
b
in
n.
Define
x=A1b.
Then
Ax=b.
The statement is false. The matrix A is invertible if and only if A is row equivalent to the identity matrix, and not every matrix A satisfying
Ax=b
is row equivalent to the identity matrix.
2.2.17
Determine whether the statement below is true or false. Justify the answer.
If A is an invertible
n×n
matrix, then the equation
Ax=b
is consistent for each
b
in
n.
Choose the correct answer below.
The statement is true. Since A is invertible,
A1b=x
for all
x
in
n.
Multiply both sides by A and the result is
Ax=b.
The statement is false. The matrix A is invertible if and only if A is row equivalent to the identity matrix, and not every matrix A satisfying
Ax=b
is row equivalent to the identity matrix.
The statement is true. Since A is invertible,
A1b
exists for all
b
in
n.
Define
x=A1b.
Then
Ax=b.
The statement is false. The matrix A satisfies
Ax=b
if and only if A is row equivalent to the identity matrix, and not every matrix that is row equivalent to the identity matrix is invertible.
2.2.18
Determine whether the statement below is true or false. Justify the answer.
If A can be row reduced to the identity matrix, then A must be invertible.
Choose the correct answer below.
The statement is false. Since the identity matrix is not invertible, A is not invertible either.
The statement is true. Since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible.
The statement is false. Not every matrix that is row equivalent to the identity matrix is invertible.
The statement is true. Since A can be row reduced to the identity matrix, I is the inverse of A. Therefore, A is invertible.
2.2.18
Determine whether the statement below is true or false. Justify the answer.
If A can be row reduced to the identity matrix, then A must be invertible.
Choose the correct answer below.
The statement is true. Since A can be row reduced to the identity matrix, A is row equivalent to the identity matrix. Since every matrix that is row equivalent to the identity is invertible, A is invertible.
The statement is false. Not every matrix that is row equivalent to the identity matrix is invertible.
The statement is false. Since the identity matrix is not invertible, A is not invertible either.
The statement is true. Since A can be row reduced to the identity matrix, I is the inverse of A. Therefore, A is invertible.
2.2.39
Find the inverse of the matrix, if it exists. Use the algorithm for finding
A1
by row reducing
[A I].
A=
16
423
Set up the matrix
[A I].
[A I]=
1610
42301
(Type an integer or simplified fraction for each matrix element.)
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
236
41
.
(Type an integer or simplified fraction for each matrix element.)
The matrix A does not have an inverse.
2.2.39
Find the inverse of the matrix, if it exists. Use the algorithm for finding
A1
by row reducing
[A I].
A=
13
411
Set up the matrix
[A I].
[A I]=
1310
41101
(Type an integer or simplified fraction for each matrix element.)
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
113
41
.
(Type an integer or simplified fraction for each matrix element.)
The matrix A does not have an inverse.
2.2.40
Find the inverse of the matrix, if it exists. Use the algorithm for finding
A1
by row reducing
[A  I].
A=
43
35
Set up the matrix
[A  I].
[A  I]=
4310
3501
(Type an integer or simplified fraction for each matrix element.)
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
511311
311411
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
2.2.40
Find the inverse of the matrix, if it exists. Use the algorithm for finding
A1
by row reducing
[A  I].
A=
63
47
Set up the matrix
[A  I].
[A  I]=
6310
4701
(Type an integer or simplified fraction for each matrix element.)
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
730110
21515
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
2.2.41
Find the inverse of the given matrix, if it exists. Use the algorithm for finding
A1
by row reducing
[A I].
A=
102
313
443
Set up the matrix
[A I].
[A I]=
102100
313010
443001
(Type an integer or simplified fraction for each matrix element.)
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
925825225
3251125925
825425125
.
(Type an integer or simplified fraction for each matrix element.)
The matrix A does not have an inverse.
2.2.41
Find the inverse of the given matrix, if it exists. Use the algorithm for finding
A1
by row reducing
[A I].
A=
103
214
424
Set up the matrix
[A I].
[A I]=
103100
214010
424001
(Type an integer or simplified fraction for each matrix element.)
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
17314328
6727514
27114128
.
(Type an integer or simplified fraction for each matrix element.)
The matrix A does not have an inverse.
2.2.41
Find the inverse of the given matrix, if it exists. Use the algorithm for finding
A1
by row reducing
[A I].
A=
103
213
243
Set up the matrix
[A I].
[A I]=
103100
213010
243001
(Type an integer or simplified fraction for each matrix element.)
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
541
431
24313
.
(Type an integer or simplified fraction for each matrix element.)
The matrix A does not have an inverse.
2.2.41
Find the inverse of the given matrix, if it exists. Use the algorithm for finding
A1
by row reducing
[A I].
A=
103
212
442
Set up the matrix
[A I].
[A I]=
103100
212010
442001
(Type an integer or simplified fraction for each matrix element.)
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
132316
297949
2929118
.
(Type an integer or simplified fraction for each matrix element.)
The matrix A does not have an inverse.
2.2.42
Find the inverse of the given matrix, if it exists.
A=
121
231
462
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A1=
(Type an integer or decimal for each matrix element.)
The matrix A does not have an inverse.
2.2.42
Find the inverse of the given matrix, if it exists.
A=
121
495
264
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A1=
(Type an integer or decimal for each matrix element.)
The matrix A does not have an inverse.
2.2.42
Find the inverse of the given matrix, if it exists.
A=
121
231
422
Find the inverse. Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
A1=
(Type an integer or decimal for each matrix element.)
The matrix A does not have an inverse.
2.2.47
Let
A=
15
16
111
.
Construct a
2×3
matrix C (by trial and error) using only 1,
1,
and 0 as entries, such that
CA=I2.
Compute AC and note that
ACI3.
C=
111
110
AC=
461
571
10121
(Simplify your answer.)
2.3.3
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
200
360
753
Choose the correct answer below.
The matrix is not invertible. If the given matrix is A, the equation
Ax=b
has no solution for some b in
3.
The matrix is not invertible. The given matrix has two pivot positions.
The matrix is invertible. The given matrix has three pivot positions.
The matrix is invertible. If the given matrix is A, there is a
3×3
matrix C such that
CI=A.
2.3.4
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
303
203
308
Choose the correct answer below.
The matrix is invertible. The given matrix has 2 pivot positions.
The matrix is not invertible. If the given matrix is A, the equation
Ax=0
has only the trivial solution.
The matrix is invertible. The columns of the given matrix span
3.
The matrix is not invertible. If the given matrix is A, the columns of A do not form a linearly independent set.
2.3.4
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
404
305
307
Choose the correct answer below.
The matrix is invertible. The columns of the given matrix span
3.
The matrix is not invertible. If the given matrix is A, the equation
Ax=0
has only the trivial solution.
The matrix is not invertible. If the given matrix is A, the columns of A do not form a linearly independent set.
The matrix is invertible. The given matrix has 2 pivot positions.
2.3.4
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
405
307
408
Choose the correct answer below.
The matrix is invertible. The given matrix has 2 pivot positions.
The matrix is invertible. The columns of the given matrix span
3.
The matrix is not invertible. If the given matrix is A, the columns of A do not form a linearly independent set.
The matrix is not invertible. If the given matrix is A, the equation
Ax=0
has only the trivial solution.
2.3.5
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
045
103
381
Choose the correct answer below.
The matrix is not invertible. If the given matrix is A, the equation
Ax=b
has a solution for at least one b in
3.
The matrix is not invertible. If the given matrix is A, A is not row equivalent to the
n×n
identity matrix.
The matrix is invertible. The columns of the given matrix span are linearly dependent.
The matrix is invertible. The given matrix has 3 pivot positions.
2.3.8
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
4575
0146
0028
0001
Choose the correct answer below.
The matrix is not invertible. If the given matrix is A, the equation
Ax=b
has no solution for at least one b in
4.
The matrix is invertible. The given matrix has 4 pivot positions.
The matrix is not invertible. In the given matrix the columns do not form a linearly independent set.
The matrix is invertible. The given matrix is not row equivalent to the
n×n
identity matrix.
2.3.8
Determine if the matrix below is invertible. Use as few calculations as possible. Justify your answer.
3585
0136
0029
0001
Choose the correct answer below.
The matrix is not invertible. If the given matrix is A, the equation
Ax=b
has no solution for at least one b in
4.
The matrix is invertible. The given matrix has 4 pivot positions.
The matrix is not invertible. In the given matrix the columns do not form a linearly independent set.
The matrix is invertible. The given matrix is not row equivalent to the
n×n
identity matrix.
2.3.11
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If the equation
Ax=0
has only the trivial solution, then A is row equivalent to the
n×n
identity matrix.
Choose the correct answer below.
The statement is false. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is not invertible. Thus, A cannot be row equivalent to the
n×n
identity matrix.
The statement is true. By the Invertible Matrix Theorem, if equation
Ax=0
has only the trivial solution, then the equation
Ax=b
has no solutions for each b in
n.
Thus, A must also be row equivalent to the
n×n
identity matrix.
The statement is false. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is not invertible; this means the columns of A do not span
n.
Thus, A must also be row equivalent to the
n×n
identity matrix.
The statement is true. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the
n×n
identity matrix.
2.3.11
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If the equation
Ax=0
has only the trivial solution, then A is row equivalent to the
n×n
identity matrix.
Choose the correct answer below.
The statement is true. By the Invertible Matrix Theorem, if equation
Ax=0
has only the trivial solution, then the equation
Ax=b
has no solutions for each b in
n.
Thus, A must also be row equivalent to the
n×n
identity matrix.
The statement is false. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is not invertible. Thus, A cannot be row equivalent to the
n×n
identity matrix.
The statement is true. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the
n×n
identity matrix.
The statement is false. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is not invertible; this means the columns of A do not span
n.
Thus, A must also be row equivalent to the
n×n
identity matrix.
2.3.11
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If the equation
Ax=0
has only the trivial solution, then A is row equivalent to the
n×n
identity matrix.
Choose the correct answer below.
The statement is true. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is invertible. Thus, A must also be row equivalent to the
n×n
identity matrix.
The statement is false. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is not invertible. Thus, A cannot be row equivalent to the
n×n
identity matrix.
The statement is true. By the Invertible Matrix Theorem, if equation
Ax=0
has only the trivial solution, then the equation
Ax=b
has no solutions for each b in
n.
Thus, A must also be row equivalent to the
n×n
identity matrix.
The statement is false. By the Invertible Matrix Theorem, if the equation
Ax=0
has only the trivial solution, then the matrix is not invertible; this means the columns of A do not span
n.
Thus, A must also be row equivalent to the
n×n
identity matrix.
2.3.12
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If there is an
n×n
matrix D such that
AD=I,
then there is also an
n×n
matrix C such that
CA=I.
Choose the correct answer below.
The statement is false. It does not follow from the Invertible Matrix Theorem that if
AD=I,
then
CA=I.
The statement is true. By the Invertible Matrix Theorem, if there is an
n×n
matrix D such that
AD=I,
then it must be true that there is also an
n×n
matrix C such that
CA=I.
The statement is false. Matrix multiplication is not commutative. It is possible that only D (and not A) is invertible in the equation
AD=I.
This implies that
CA=I
is only true when C is invertible (for cases where A is not invertible), but it is not given that C is invertible.
The statement is true. By the Invertible Matrix Theorem, if
AD=I
then A, D, or both is/are the identity matrix. Therefore,
CA=I.
2.3.12
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If there is an
n×n
matrix D such that
AD=I,
then there is also an
n×n
matrix C such that
CA=I.
Choose the correct answer below.
The statement is false. Matrix multiplication is not commutative. It is possible that only D (and not A) is invertible in the equation
AD=I.
This implies that
CA=I
is only true when C is invertible (for cases where A is not invertible), but it is not given that C is invertible.
The statement is true. By the Invertible Matrix Theorem, if
AD=I
then A, D, or both is/are the identity matrix. Therefore,
CA=I.
The statement is false. It does not follow from the Invertible Matrix Theorem that if
AD=I,
then
CA=I.
The statement is true. By the Invertible Matrix Theorem, if there is an
n×n
matrix D such that
AD=I,
then it must be true that there is also an
n×n
matrix C such that
CA=I.
2.3.15
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If A is an
n×n
matrix, then the equation
Ax=b
has at least one solution for each
b
in
n.
Choose the correct answer below.
The statement is true. By the Invertible Matrix Theorem,
Ax=b
has at least one solution for each
b
in
n
for all matrices of size
n×n.
The statement is false. By the Invertible Matrix Theorem,
Ax=b
has at least one solution for each
b
in
n
only if a matrix is invertible.
The statement is true. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each
b
in
n,
then the matrix is not invertible.
The statement is false. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each
b
in
n,
then the equation
Ax=b
has no solution.
2.3.15
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If A is an
n×n
matrix, then the equation
Ax=b
has at least one solution for each
b
in
n.
Choose the correct answer below.
The statement is true. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each
b
in
n,
then the matrix is not invertible.
The statement is false. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each
b
in
n,
then the equation
Ax=b
has no solution.
The statement is true. By the Invertible Matrix Theorem,
Ax=b
has at least one solution for each
b
in
n
for all matrices of size
n×n.
The statement is false. By the Invertible Matrix Theorem,
Ax=b
has at least one solution for each
b
in
n
only if a matrix is invertible.
2.3.15
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If A is an
n×n
matrix, then the equation
Ax=b
has at least one solution for each
b
in
n.
Choose the correct answer below.
The statement is true. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each
b
in
n,
then the matrix is not invertible.
The statement is true. By the Invertible Matrix Theorem,
Ax=b
has at least one solution for each
b
in
n
for all matrices of size
n×n.
The statement is false. By the Invertible Matrix Theorem,
Ax=b
has at least one solution for each
b
in
n
only if a matrix is invertible.
The statement is false. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each
b
in
n,
then the equation
Ax=b
has no solution.
2.3.16
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If the equation
Ax=b
has at least one solution for each b in
n,
then the solution is unique for each
b.
Choose the correct answer below.
The statement is false. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each b in
n,
then matrix A is not invertible. If A is not invertible, then according to the Invertible Matrix Theorem, the solution is not unique for each
b.
The statement is true. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each b in
n,
then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each
b.
The statement is true, but only for
x0.
By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each b in
n,
then the equation
Ax=0
does not only have the trivial solution.
The statement is false. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each b in
n,
then the linear transformation
x  Ax
does not map
n
onto
n.
2.3.16
For this exercise assume that the matrices are all
n×n.
The statement in this exercise is an implication of the form "If "statement 1", then "statement 2"." Mark an implication as True if the truth of "statement 2" always follows whenever "statement 1" happens to be true. Mark the implication as False if "statement 2" is false but "statement 1" is true. Justify your answer.
If the equation
Ax=b
has at least one solution for each b in
n,
then the solution is unique for each
b.
Choose the correct answer below.
The statement is false. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each b in
n,
then the linear transformation
x  Ax
does not map
n
onto
n.
The statement is true. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each b in
n,
then matrix A is invertible. If A is invertible, then according to the invertible matrix theorem the solution is unique for each
b.
The statement is true, but only for
x0.
By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each b in
n,
then the equation
Ax=0
does not only have the trivial solution.
The statement is false. By the Invertible Matrix Theorem, if
Ax=b
has at least one solution for each b in
n,
then matrix A is not invertible. If A is not invertible, then according to the Invertible Matrix Theorem, the solution is not unique for each
b.
2.3.21
An
m×n
upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer.
3474
0146
0028
0001
Choose the correct answer below.
A square upper triangular matrix is invertible when all entries on the main diagonal are ones. If any entry on the main diagonal is not one, then the equation
Ax=b,
where A is an
n×n
square upper triangular matrix, has no solution for at least one b in
n.
A square upper triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the
n×n
matrix has n pivot positions.
A square upper triangular matrix is invertible when all entries above the main diagonal are zeros as well. This means that the matrix is row equivalent to the
n×n
identity matrix.
A square upper triangular matrix is invertible when the matrix is equal to its own transpose. For such a matrix A,
A=AT
means that the equation
Ax=b
has at least one solution for each b in
n.
2.3.21
An
m×n
upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer.
3474
0146
0028
0001
Choose the correct answer below.
A square upper triangular matrix is invertible when the matrix is equal to its own transpose. For such a matrix A,
A=AT
means that the equation
Ax=b
has at least one solution for each b in
n.
A square upper triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the
n×n
matrix has n pivot positions.
A square upper triangular matrix is invertible when all entries on the main diagonal are ones. If any entry on the main diagonal is not one, then the equation
Ax=b,
where A is an
n×n
square upper triangular matrix, has no solution for at least one b in
n.
A square upper triangular matrix is invertible when all entries above the main diagonal are zeros as well. This means that the matrix is row equivalent to the
n×n
identity matrix.
2.3.21
An
m×n
upper triangular matrix is one whose entries below the main diagonal are zeros, as is shown in the matrix to the right. When is a square upper triangular matrix invertible? Justify your answer.
3474
0146
0028
0001
Choose the correct answer below.
A square upper triangular matrix is invertible when all entries on the main diagonal are ones. If any entry on the main diagonal is not one, then the equation
Ax=b,
where A is an
n×n
square upper triangular matrix, has no solution for at least one b in
n.
A square upper triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the
n×n
matrix has n pivot positions.
A square upper triangular matrix is invertible when all entries above the main diagonal are zeros as well. This means that the matrix is row equivalent to the
n×n
identity matrix.
A square upper triangular matrix is invertible when the matrix is equal to its own transpose. For such a matrix A,
A=AT
means that the equation
Ax=b
has at least one solution for each b in
n.
2.3.22
An
m×n
lower triangular matrix is one whose entries above the main diagonal are zeros, as is shown in the matrix to the right. When is a square lower triangular matrix invertible? Justify your answer.
3000
4100
7420
4681
Choose the correct answer below.
A square lower triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the
n×n
matrix has n pivot positions.
A square lower triangular matrix is invertible when all entries below the main diagonal are zeros as well. This means that the matrix is row equivalent to the
n×n
identity matrix.
A square lower triangular matrix is invertible when all entries on the main diagonal are ones. If any entry on the main diagonal is not one, then the equation
Ax=b,
where A is an
n×n
square lower triangular matrix, has no solution for at least one b in
n.
A square lower triangular matrix is invertible when the matrix is equal to its own transpose. For such a matrix A,
A=AT
means that the equation
Ax=b
has at least one solution for each b in
n.
2.3.22
An
m×n
lower triangular matrix is one whose entries above the main diagonal are zeros, as is shown in the matrix to the right. When is a square lower triangular matrix invertible? Justify your answer.
3000
4100
7420
4681
Choose the correct answer below.
A square lower triangular matrix is invertible when the matrix is equal to its own transpose. For such a matrix A,
A=AT
means that the equation
Ax=b
has at least one solution for each b in
n.
A square lower triangular matrix is invertible when all entries on its main diagonal are nonzero. If all of the entries on its main diagonal are nonzero, then the
n×n
matrix has n pivot positions.
A square lower triangular matrix is invertible when all entries below the main diagonal are zeros as well. This means that the matrix is row equivalent to the
n×n
identity matrix.
A square lower triangular matrix is invertible when all entries on the main diagonal are ones. If any entry on the main diagonal is not one, then the equation
Ax=b,
where A is an
n×n
square lower triangular matrix, has no solution for at least one b in
n.
2.3.42
The given T is a linear transformation from
2
into
2.
Show that T is invertible and find a formula for
T1.
Tx1,x2=4x16x2,4x1+9x2
To show that T is invertible, calculate the determinant of the standard matrix for T. The determinant of the standard matrix is
12.
(Simplify your answer.)
T1x1,x2=34x1+12x2,13x1+13x2
(Type an ordered pair. Type an expression using
x1
and
x2
as the variables.)
2.8.1
A set in
2
is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of
2.
(For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)
Let u and v be vectors and let k be a scalar. Select the correct choice below and, if necessary, fill in the answer box within your choice.
The set is not a subspace because it does not include the zero vector.
0
The set is not a subspace because it is not closed under either scalar multiplication or sums. For example,
multiplied by (1,3) is not in the set, and the sum of (3,1) and (1,3) is not in the set.
u
v
u+v
ku
The set is not a subspace because it is closed under scalar multiplication, but not under sums. For example, the sum of (3,1) and (1,3) is not in the set.
u
v
u+v
The set is not a subspace because it is closed under sums, but not under scalar multiplication. For example,
1
multiplied by (1,1) is not in the set.
u
ku
2.8.1
A set in
2
is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of
2.
(For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)
Let u and v be vectors and let k be a scalar. Select the correct choice below and, if necessary, fill in the answer box within your choice.
The set is not a subspace because it does not include the zero vector.
0
The set is not a subspace because it is closed under sums, but not under scalar multiplication. For example,
1
multiplied by (1,1) is not in the set.
u
ku
The set is not a subspace because it is not closed under either scalar multiplication or sums. For example,
multiplied by (1,3) is not in the set, and the sum of (3,1) and (1,3) is not in the set.
u
v
u+v
ku
The set is not a subspace because it is closed under scalar multiplication, but not under sums. For example, the sum of (3,1) and (1,3) is not in the set.
u
v
u+v
2.8.2
A set in
2
is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of
2.
(For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)
Let u and v be vectors and let k be a scalar. Select the correct choice below and, if necessary, fill in the answer box within your choice.
The set is not a subspace because it is closed under scalar multiplication, but not under sums. For example, the sum of (2,2) and
(1,3)
is not in the set.
u
v
u+v
The set is not a subspace because it does not include the zero vector.
0
The set is not a subspace because it is closed under sums, but not under scalar multiplication. For example,
multiplied by (0,1) is not in the set.
u
ku
The set is not a subspace because it is not closed under either scalar multiplication or sums. For example,
multiplied by (0,1) is not in the set, and the sum of (2,2) and
(1,3)
is not in the set.
u
v
u+v
ku
2.8.2
A set in
2
is displayed to the right. Assume the set includes the bounding lines. Give a specific reason why the set H is not a subspace of
2.
(For instance, find two vectors in H whose sum is not in H, or find a vector in H with a scalar multiple that is not in H. Draw a picture.)
Let u and v be vectors and let k be a scalar. Select the correct choice below and, if necessary, fill in the answer box within your choice.
The set is not a subspace because it is closed under sums, but not under scalar multiplication. For example,
multiplied by (0,1) is not in the set.
u
ku
The set is not a subspace because it is not closed under either scalar multiplication or sums. For example,
multiplied by (0,1) is not in the set, and the sum of (2,2) and
(1,3)
is not in the set.
u
v
u+v
ku
The set is not a subspace because it is closed under scalar multiplication, but not under sums. For example, the sum of (2,2) and
(1,3)
is not in the set.
u
v
u+v
The set is not a subspace because it does not include the zero vector.
0
2.8.5
Let
v1=
1
3
5
,
v2=
2
3
9
,
and
w=
3
3
13
.
Determine if
w
is
in the subspace of
3
generated by
v1
and
v2.
Is
w
is
in the subspace of
3
generated by
v1
and
v2?
Yes
No
2.8.5
Let
v1=
1
3
4
,
v2=
2
3
7
,
and
w=
3
3
10
.
Determine if
w
is
in the subspace of
3
generated by
v1
and
v2.
Is
w
is
in the subspace of
3
generated by
v1
and
v2?
No
Yes
2.8.5
Let
v1=
1
3
5
,
v2=
2
2
9
,
and
w=
3
1
13
.
Determine if
w
is
in the subspace of
3
generated by
v1
and
v2.
Is
w
is
in the subspace of
3
generated by
v1
and
v2?
No
Yes
2.8.7
Let
v1=
1
3
0
,
v2=
0
1
4
,
v3=
4
6
24
,
p=
2
3
12
,
and
A=
v1v2v3
.
a. How many vectors are in
{v1,
v2,
v3}?
b. How many vectors are in Col A?
c. Is
p
in Col A? Why or why not?
a. How many vectors are in
{v1,
v2,
v3}?
Select the correct choice below and, if necessary, fill in the answer box within your choice.
3
(Type a whole number.)
There are infinitely many vectors in
{v1,
v2,
v3}.
b. How many vectors are in Col A? Select the correct choice below and, if necessary, fill in the answer box within your choice.
(Type a whole number.)
There are infinitely many vectors in Col A.
c. Is
p
in Col A? Why or why not?
p
is in Col A because A has pivot positions in every row.
p
is in Col A because the system
Ap
is consistent.
p
is not in Col A because A has too few pivot positions.
p
is not in Col A because the system
Ap
is not consistent.
2.8.7
Let
v1=
1
2
0
,
v2=
0
1
2
,
v3=
4
5
6
,
p=
4
2
12
,
and
A=
v1v2v3
.
a. How many vectors are in
{v1,
v2,
v3}?
b. How many vectors are in Col A?
c. Is
p
in Col A? Why or why not?
a. How many vectors are in
{v1,
v2,
v3}?
Select the correct choice below and, if necessary, fill in the answer box within your choice.
3
(Type a whole number.)
There are infinitely many vectors in
{v1,
v2,
v3}.
b. How many vectors are in Col A? Select the correct choice below and, if necessary, fill in the answer box within your choice.
(Type a whole number.)
There are infinitely many vectors in Col A.
c. Is
p
in Col A? Why or why not?
p
is in Col A because the system
Ap
is consistent.
p
is not in Col A because A has too few pivot positions.
p
is not in Col A because the system
Ap
is not consistent.
p
is in Col A because A has pivot positions in every row.
2.8.9
Let
v1=
9
0
18
,
v2=
6
6
9
,
v3=
0
18
9
,
and
p=
2
10
7
.
Determine if
p
is in Nul A, where
A=
v1v2v3
.
Is
p
in Nul A?
No, because
Ap
is not equal to the zero vector.
Yes, because
Ap
is equal to the zero vector.
No, because the augmented matrix
Ap
is not consistent.
Yes, because the augmented matrix
Ap
is consistent.
2.8.9
Let
v1=
3
0
6
,
v2=
2
2
3
,
v3=
0
6
3
,
and
p=
5
6
8
.
Determine if
p
is in Nul A, where
A=
v1v2v3
.
Is
p
in Nul A?
No, because
Ap
is not equal to the zero vector.
No, because the augmented matrix
Ap
is not consistent.
Yes, because the augmented matrix
Ap
is consistent.
Yes, because
Ap
is equal to the zero vector.
2.8.9
Let
v1=
6
0
12
,
v2=
4
4
6
,
v3=
0
12
6
,
and
p=
5
8
1
.
Determine if
p
is in Nul A, where
A=
v1v2v3
.
Is
p
in Nul A?
No, because
Ap
is not equal to the zero vector.
No, because the augmented matrix
Ap
is not consistent.
Yes, because
Ap
is equal to the zero vector.
Yes, because the augmented matrix
Ap
is consistent.
2.8.10
Let
v1=
4
1
5
,
v2=
4
5
5
,
v3=
1
7
7
,
and
u=
7
7
5
.
Determine if u is in Nul A, where
A=
v1v2v3
.
Is u in Nul A?
No
Yes
2.8.10
Let
v1=
3
0
4
,
v2=
3
2
2
,
v3=
0
7
7
,
and
u=
7
7
2
.
Determine if u is in Nul A, where
A=
v1v2v3
.
Is u in Nul A?
No
Yes
2.8.10
Let
v1=
3
0
12
,
v2=
3
6
6
,
v3=
0
8
8
,
and
u=
8
8
6
.
Determine if u is in Nul A, where
A=
v1v2v3
.
Is u in Nul A?
Yes
No
2.8.11
Give integers p and q such that
Nul A
is a subspace of
p
and
Col A
is a subspace of
q.
A=
123
457
Nul A
is a subspace of
p
for
p=3
and
Col A
is a subspace of
q
for
q=2.
2.8.11
Give integers p and q such that
Nul A
is a subspace of
p
and
Col A
is a subspace of
q.
A=
54321
03408
90661
Nul A
is a subspace of
p
for
p=5
and
Col A
is a subspace of
q
for
q=3.
2.8.13
For the matrix A below, find a nonzero vector in Nul A and a nonzero vector in Col A.
A=
3212
126120
902718
Find a nonzero vector in Nul A.
1
0
1
1
Find a nonzero vector in Col A.
3
12
9
2.8.15
Determine if the set is a basis for
2.
Justify your answer.
16
4
,
64
2
Is the given set a basis for
2?
No, because these vectors form the columns of an invertible
2×2
matrix. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes, because these vectors form the columns of an invertible
2×2
matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes, because these vectors form the columns of a
2×2
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No, because these vectors form the columns of a
2×2
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
2.8.15
Determine if the set is a basis for
2.
Justify your answer.
8
2
,
32
9
Is the given set a basis for
2?
No, because these vectors form the columns of a
2×2
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No, because these vectors form the columns of an invertible
2×2
matrix. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes, because these vectors form the columns of an invertible
2×2
matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes, because these vectors form the columns of a
2×2
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
2.8.17
Determine if the set is a basis for
3.
Justify your answer.
0
0
4
,
3
6
8
,
4
8
4
Is the given set a basis for
3?
No, because these three vectors form the columns of an invertible
3×3
matrix. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes,
because these three vectors form the columns of an invertible 3×3 matrix.
By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes, because these three vectors form the columns of a
3×3
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No,
because these three vectors form the columns of a 3×3 matrix that is not invertible.
By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
2.8.17
Determine if the set is a basis for
3.
Justify your answer.
0
0
2
,
1
2
4
,
2
4
2
Is the given set a basis for
3?
Yes, because these three vectors form the columns of a
3×3
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes,
because these three vectors form the columns of an invertible 3×3 matrix.
By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No,
because these three vectors form the columns of a 3×3 matrix that is not invertible.
By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No, because these three vectors form the columns of an invertible
3×3
matrix. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
2.8.17
Determine if the set is a basis for
3.
Justify your answer.
0
0
2
,
7
0
4
,
3
3
2
Is the given set a basis for
3?
Yes, because these three vectors form the columns of a
3×3
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes,
because these three vectors form the columns of an invertible 3×3 matrix.
By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No,
because these three vectors form the columns of a 3×3 matrix that is not invertible.
By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No, because these three vectors form the columns of an invertible
3×3
matrix. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
2.8.19
Determine if the set is a basis for
3.
Justify your answer.
3
4
1
,
6
1
4
Is the given set a basis for
3?
Yes, because these vectors form the columns of an invertible
3×3
matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No, because these two vectors are linearly dependent.
No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of
3.
Yes, because these two vectors are linearly independent.
2.8.19
Determine if the set is a basis for
3.
Justify your answer.
4
4
1
,
8
1
4
Is the given set a basis for
3?
No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of
3.
Yes, because these two vectors are linearly independent.
Yes, because these vectors form the columns of an invertible
3×3
matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No, because these two vectors are linearly dependent.
2.8.19
Determine if the set is a basis for
3.
Justify your answer.
2
8
1
,
4
2
5
Is the given set a basis for
3?
No, because these vectors form a matrix with only 2 pivot columns. Therefore, these vectors form a basis for a two-dimensional subspace of
3.
No, because these two vectors are linearly dependent.
Yes, because these two vectors are linearly independent.
Yes, because these vectors form the columns of an invertible
3×3
matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
2.8.20
Determine if the set is a basis for
3.
Justify your answer.
1
5
2
,
2
3
3
,
3
3
11
,
0
7
5
Is the given set a basis for
3?
No, because these vectors form the columns of a
3×3
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No, because these vectors do not form the columns of a
3×3
matrix. A set that contains more vectors than there are entries is linearly dependent.
Yes, because these vectors form the columns of an invertible
3×3
matrix. A set that contains more vectors than there are entries is linearly independent.
Yes, because these vectors form the columns of an invertible
3×3
matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
2.8.20
Determine if the set is a basis for
3.
Justify your answer.
1
5
2
,
7
9
5
,
7
7
11
,
0
3
7
Is the given set a basis for
3?
Yes, because these vectors form the columns of an invertible
3×3
matrix. By the invertible matrix theorem, the following statements are equivalent: A is an invertible matrix, the columns of A form a linearly independent set, and the columns of A span
n.
No, because these vectors do not form the columns of a
3×3
matrix. A set that contains more vectors than there are entries is linearly dependent.
No, because these vectors form the columns of a
3×3
matrix that is not invertible. By the invertible matrix theorem, the following statements are equivalent: A is a singular matrix, the columns of A form a linearly independent set, and the columns of A span
n.
Yes, because these vectors form the columns of an invertible
3×3
matrix. A set that contains more vectors than there are entries is linearly independent.
2.8.21
Determine whether the statement below is true or false. Justify the answer.
A subspace of
n
is any set H such that (i) the zero vector is in H, (ii)
u,
v,
and
u+v
are in H, and (iii) c is a scalar and
cu
is in H.
Choose the correct answer below.
The statement is false. It must also be separately specified that
uv
are in H when
u
and
v
are in H.
The statement is true. Any set of elements of
n
that satisfies conditions (i), (ii), and (iii) is a subspace by definition.
The statement is false. The zero vector does not need to be in a set for the set to be a subspace.
The statement is false. Conditions (ii) and (iii) must be satisfied for each
u
and
v
in H, which is not specified in the given statement.
The statement is false. It must also be separately specified that
au+bv
are in H when
u
and
v
are in H and a and b are scalars.
2.8.22
Determine whether the statement below is true or false. Justify the answer.
A subset H of
n
is a subspace if the zero vector is in H.
Choose the correct answer below.
This statement is false. For each u and v in H, the product uv must also be in H.
This statement is false. The subset H is a subspace if the zero vector is not in H.
This statement is false. For each u and v in H and each scalar c, the sum
u+v
and the vector cu must also be in H.
This statement is true. This is the definition of a subspace.
2.8.23
Determine whether the statement below is true or false. Justify the answer.
If
v1, ..., vp
are in
n,
then
S=Spanv1, ..., vp
is the same as the column space of the matrix
A=v1     vp.
Choose the correct answer below.
The statement is true. The vectors
v1, ..., vp
always form a basis of both S and the column space of A.
The statement is true. The column space of A and S are both the set of the vectors
v1, ..., vp
and the zero vector.
The statement is true. The column space of A and S are both the set of all linear combinations of
v1, ..., vp.
The statement is false. There are infinitely many vectors in S but finitely many vectors in the column space of A.
The statement is false. There are finitely many vectors in S but infinitely many vectors in the column space of A.
The statement is false. S is the same as the column space of A only if the columns of A are linearly independent.
2.8.23
Determine whether the statement below is true or false. Justify the answer.
If
v1, ..., vp
are in
n,
then
S=Spanv1, ..., vp
is the same as the column space of the matrix
A=v1     vp.
Choose the correct answer below.
The statement is true. The column space of A and S are both the set of all linear combinations of
v1, ..., vp.
The statement is true. The vectors
v1, ..., vp
always form a basis of both S and the column space of A.
The statement is true. The column space of A and S are both the set of the vectors
v1, ..., vp
and the zero vector.
The statement is false. S is the same as the column space of A only if the columns of A are linearly independent.
The statement is false. There are infinitely many vectors in S but finitely many vectors in the column space of A.
The statement is false. There are finitely many vectors in S but infinitely many vectors in the column space of A.
2.8.24
Determine whether the statement below is true or false. Justify the answer.
Given vectors
v1,
...,
vp
in
n,
the set of all linear combinations of these vectors is a subspace of
n.
Choose the correct answer below.
This statement is false. This set is a subspace of
p.
This statement is true. This set satisfies all properties of a subspace.
This statement is false. This set does not contain the zero vector.
This statement is false. This set is a subspace of
n+p.
2.8.24
Determine whether the statement below is true or false. Justify the answer.
Given vectors
v1,
...,
vp
in
n,
the set of all linear combinations of these vectors is a subspace of
n.
Choose the correct answer below.
This statement is false. This set does not contain the zero vector.
This statement is false. This set is a subspace of
p.
This statement is false. This set is a subspace of
n+p.
This statement is true. This set satisfies all properties of a subspace.
2.8.24
Determine whether the statement below is true or false. Justify the answer.
Given vectors
v1,
...,
vp
in
n,
the set of all linear combinations of these vectors is a subspace of
n.
Choose the correct answer below.
This statement is false. This set is a subspace of
n+p.
This statement is false. This set does not contain the zero vector.
This statement is false. This set is a subspace of
p.
This statement is true. This set satisfies all properties of a subspace.
2.8.25
Determine whether the statement below is true or false. Justify the answer.
The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of
m.
Choose the correct answer below.
The statement is true. The described set is the null space of an
m×n
matrix A. This set is a subspace of
m.
The statement is false. The described set is the column space of an
m×n
matrix A. This set is a subspace of
n.
The statement is false. The described set is the null space of an
m×n
matrix A. This set is a subspace of
n.
The statement is false. The described set is only a subspace of
m
if
m<n.
The statement is true. The described set is the column space of an
m×n
matrix A. This set is a subspace of
m.
2.8.25
Determine whether the statement below is true or false. Justify the answer.
The set of all solutions of a system of m homogeneous equations in n unknowns is a subspace of
m.
Choose the correct answer below.
The statement is false. The described set is only a subspace of
m
if
m<n.
The statement is false. The described set is the null space of an
m×n
matrix A. This set is a subspace of
n.
The statement is true. The described set is the null space of an
m×n
matrix A. This set is a subspace of
m.
The statement is false. The described set is the column space of an
m×n
matrix A. This set is a subspace of
n.
The statement is true. The described set is the column space of an
m×n
matrix A. This set is a subspace of
m.
2.8.26
Determine whether the statement below is true or false. Justify the answer.
The null space of an
m×n
matrix is a subspace of
n.
Choose the correct answer below.
This statement is false. This set is not closed under scalar multiplication.
This statement is true. For an
m×n
matrix A, the solutions of
Ax=0
are vectors in
n
and satisfy the properties of a vector space.
This statement is false. For an
m×n
matrix A, the solutions of
Ax=0
belong to
m.
This statement is false. The null space of a matrix does not contain the zero vector.
2.8.27
Determine whether the statement below is true or false. Justify the answer.
The columns of an invertible
n×n
matrix form a basis for
n.
Choose the correct answer below.
The statement is false. There are not enough columns in an
n×n
matrix to form a basis for
n.
The statement is true. The columns of an invertible
n×n
matrix are linearly independent and span
n,
so they form a basis for
n.
The statement is false. The columns of an invertible
n×n
matrix are not linearly independent. This means they cannot form a basis for any vector space, including
n.
The statement is false. The columns of an invertible
n×n
matrix are linearly independent, but they do not span
n.
This means they cannot form a basis for
n.
The statement is true. There are n columns in an
n×n
matrix, each of which is a vector in
n.
Any set of n vectors in
n
must form a basis for
n.
2.8.27
Determine whether the statement below is true or false. Justify the answer.
The columns of an invertible
n×n
matrix form a basis for
n.
Choose the correct answer below.
The statement is true. The columns of an invertible
n×n
matrix are linearly independent and span
n,
so they form a basis for
n.
The statement is false. The columns of an invertible
n×n
matrix are not linearly independent. This means they cannot form a basis for any vector space, including
n.
The statement is true. There are n columns in an
n×n
matrix, each of which is a vector in
n.
Any set of n vectors in
n
must form a basis for
n.
The statement is false. There are not enough columns in an
n×n
matrix to form a basis for
n.
The statement is false. The columns of an invertible
n×n
matrix are linearly independent, but they do not span
n.
This means they cannot form a basis for
n.
2.8.31
A matrix A and an echelon form of A are shown below. Find a basis for Col A and a basis for Nul A.
A
=
581739
5279
2122
~
12511
0148
0000
Find a basis for Col A.
5
5
2
,
8
2
1
(Simplify your answer. Use a comma to separate answers as needed.)
Find a basis for Nul A.
3
4
1
0
,
5
8
0
1
(Simplify your answer. Use a comma to separate answers as needed.)
2.8.31
A matrix A and an echelon form of A are shown below. Find a basis for Col A and a basis for Nul A.
A
=
362115
7155440
383125
~
131210
0155
0000
Find a basis for Col A.
3
7
3
,
6
15
8
(Simplify your answer. Use a comma to separate answers as needed.)
Find a basis for Nul A.
3
5
1
0
,
5
5
0
1
(Simplify your answer. Use a comma to separate answers as needed.)
2.8.31
A matrix A and an echelon form of A are shown below. Find a basis for Col A and a basis for Nul A.
A
=
371938
6102244
371938
~
13918
0148
0000
Find a basis for Col A.
3
6
3
,
7
10
7
(Simplify your answer. Use a comma to separate answers as needed.)
Find a basis for Nul A.
3
4
1
0
,
6
8
0
1
(Simplify your answer. Use a comma to separate answers as needed.)
2.8.33
A matrix A and an echelon form of A are shown below. Find a basis for Col A and a basis for Nul A.
A
=
1152511
1152947
25564
3152169
~
1101605
05902
00012
00000
Find a basis for Col A.
1
1
2
3
,
15
15
5
15
,
1
4
6
6
(Use a comma to separate answers as needed. Type an integer or simplified fraction for each matrix element.)
Find a basis for Nul A.
2
95
1
0
0
,
9
25
0
2
1
(Use a comma to separate answers as needed. Type an integer or simplified fraction for each matrix element.)
2.9.3
The vector x is in a subspace H with a basis
B={b1,b2}.
Find the B-coordinate vector of
x.
b1=
2
3
,
b2=
1
4
,
x=
6
4
[x]B=
4
2
2.9.3
The vector x is in a subspace H with a basis
B={b1,b2}.
Find the B-coordinate vector of
x.
b1=
2
3
,
b2=
1
3
,
x=
6
6
[x]B=
4
2
2.9.3
The vector x is in a subspace H with a basis
B={b1,b2}.
Find the B-coordinate vector of
x.
b1=
2
5
,
b2=
1
4
,
x=
4
7
[x]B=
3
2
2.9.5
The vector x is in a subspace H with a basis
B={b1,b2}.
Find the B-coordinate vector of
x.
b1=
1
2
4
,
b2=
4
7
15
,
x=
7
12
26
[x]B=
1
2
2.9.5
The vector x is in a subspace H with a basis
B={b1,b2}.
Find the B-coordinate vector of
x.
b1=
1
2
3
,
b2=
4
7
11
,
x=
11
18
29
[x]B=
5
4
2.9.5
The vector x is in a subspace H with a basis
B={b1,b2}.
Find the B-coordinate vector of
x.
b1=
1
4
3
,
b2=
3
11
8
,
x=
5
17
12
[x]B=
4
3
2.9.9
Given below is a matrix A and an echelon form of A. Find bases for
Col A
and
Nul A,
and then state the dimensions of these subspaces.
A=
1354
51513
3967
412013
~
1336
0078
0007
0000
A basis for
Col A
is given by
1
5
3
4
,
5
1
6
0
,
4
3
7
13
.
(Use a comma to separate answers as needed.)
The dimension of
Col A
is
3.
(Type an integer.)
A basis for
Nul A
is given by
3
1
0
0
.
(Use a comma to separate answers as needed.)
The dimension of
Nul A
is
1.
(Type an integer.)
2.9.9
Given below is a matrix A and an echelon form of A. Find bases for
Col A
and
Nul A,
and then state the dimensions of these subspaces.
A=
1256
3614
612510
48012
~
1263
0038
0003
0000
A basis for
Col A
is given by
1
3
6
4
,
5
1
5
0
,
6
4
10
12
.
(Use a comma to separate answers as needed.)
The dimension of
Col A
is
3.
(Type an integer.)
A basis for
Nul A
is given by
2
1
0
0
.
(Use a comma to separate answers as needed.)
The dimension of
Nul A
is
1.
(Type an integer.)
2.9.9
Given below is a matrix A and an echelon form of A. Find bases for
Col A
and
Nul A,
and then state the dimensions of these subspaces.
A=
1237
4815
36612
612019
~
1234
0049
0004
0000
A basis for
Col A
is given by
1
4
3
6
,
3
1
6
0
,
7
5
12
19
.
(Use a comma to separate answers as needed.)
The dimension of
Col A
is
3.
(Type an integer.)
A basis for
Nul A
is given by
2
1
0
0
.
(Use a comma to separate answers as needed.)
The dimension of
Nul A
is
1.
(Type an integer.)
2.9.11
Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A and an echelon form of A below.
A=
12117
25016
393117
310549
~
12117
01238
00014
00000
A basis for Col A is given by
1
2
3
3
,
2
5
9
10
,
1
1
1
4
.
(Use a comma to separate vectors as needed.)
The dimension of Col A is
3.
A basis for Nul A is given by
5
2
1
0
0
,
5
4
0
4
1
.
(Use a comma to separate vectors as needed.)
The dimension of Nul A is
2.
2.9.11
Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A and an echelon form of A below.
A=
12304
251030
392148
3102570
~
12304
01438
00014
00000
A basis for Col A is given by
1
2
3
3
,
2
5
9
10
,
0
3
4
7
.
(Use a comma to separate vectors as needed.)
The dimension of Col A is
3.
A basis for Nul A is given by
5
4
1
0
0
,
4
4
0
4
1
.
(Use a comma to separate vectors as needed.)
The dimension of Nul A is
2.
2.9.11
Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A and an echelon form of A below.
A=
13320
271063
3122176
3132593
~
13320
01423
00013
00000
A basis for Col A is given by
1
2
3
3
,
3
7
12
13
,
2
6
7
9
.
(Use a comma to separate vectors as needed.)
The dimension of Col A is
3.
A basis for Nul A is given by
9
4
1
0
0
,
3
3
0
3
1
.
(Use a comma to separate vectors as needed.)
The dimension of Nul A is
2.
2.9.12
Find the bases for Col A and Nul A, and then state the dimension of these subspaces for the matrix A and an echelon form of A below.
A=
12364
5104313
48538
24338
~
12364
00130
00008
00000
A basis for Col A is given by
1
5
4
2
,
3
4
5
3
,
4
13
8
8
.
(Use a comma to separate vectors as needed.)
The dimension of Col A is
3.
A basis for Nul A is given by
2
1
0
0
0
,
3
0
3
1
0
.
(Use a comma to separate vectors as needed.)
The dimension of Nul A is
2.
2.9.13
Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace?
1
3
4
6
,
3
9
12
18
,
2
1
5
6
,
5
7
2
12
A basis for the subspace is given by
1
3
4
6
,
2
1
5
6
,
5
7
2
12
.
(Use a comma to separate answers as needed.)
The dimension of this subspace is
3.
(Type an integer.)
2.9.13
Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace?
1
5
6
2
,
3
15
18
6
,
2
1
6
3
,
5
3
3
9
A basis for the subspace is given by
1
5
6
2
,
2
1
6
3
,
5
3
3
9
.
(Use a comma to separate answers as needed.)
The dimension of this subspace is
3.
(Type an integer.)
2.9.13
Find a basis for the subspace spanned by the given vectors. What is the dimension of the subspace?
1
6
3
2
,
2
12
6
4
,
3
1
4
5
,
8
5
6
10
A basis for the subspace is given by
1
6
3
2
,
3
1
4
5
,
8
5
6
10
.
(Use a comma to separate answers as needed.)
The dimension of this subspace is
3.
(Type an integer.)
2.9.15
Suppose a
3×5
matrix A has
three
pivot columns. Is Col
A=3?
Is Nul
A=2?
Explain your answers.
Is Col
A=3?
Explain your answer. Choose the correct answer and reasoning below.
No, because a
3×5
matrix exists in
5.
If its pivot columns form a
3-dimensional
basis, then Col A is isomorphic to
3
but is not strictly equal to
3.
Yes, because there are
three
pivot columns in A. These columns form a basis in
three
dimensions. Any
3-dimensional
basis spans
3.
Yes, because the column space of a
3×5
matrix is a subspace of
3.
There is a pivot in each row, so the column space is
3-dimensional.
Since any
3-dimensional
subspace of
3
is
3,
Col
A=3.
No, Col
A=2.
The number of pivot columns is equal to the dimension of the null space. Since the sum of the dimensions of the null space and column space equals the number of columns in the matrix, the dimension of the column space must be
2.
Since any
2-dimensional
basis is equal to
2,
Col
A=2.
Is Nul
A=2?
Explain your answer. Choose the correct answer and reasoning below.
Yes, because the linearly dependent vectors in A form a basis in
two
dimensions. Any basis in
two
dimensions is also a basis for
2.
Therefore, Nul
A=2.
Yes, because a
3×5
matrix exists in
3.
Therefore, if its null space is
2-dimensional
and contained within
3,
it must be equal to
2.
No, because the null space of a
3×5
matrix is a subspace of
5.
Although
dim Nul A=2,
it is not strictly equal to
2
because each vector in Nul A has
five
components. Each vector in
2
has
two
components. Therefore, Nul A is isomorphic to
2,
but not equal.
No, because although the null space is
2-dimensional,
its basis consists of
three
vectors and not
two.
Therefore, it cannot be equal to
2.
2.9.15
Suppose a
3×6
matrix A has
three
pivot columns. Is Col
A=3?
Is Nul
A=3?
Explain your answers.
Is Col
A=3?
Explain your answer. Choose the correct answer and reasoning below.
No, Col
A=3.
The number of pivot columns is equal to the dimension of the null space. Since the sum of the dimensions of the null space and column space equals the number of columns in the matrix, the dimension of the column space must be
3.
Since any
3-dimensional
basis is equal to
3,
Col
A=3.
No, because a
3×6
matrix exists in
6.
If its pivot columns form a
3-dimensional
basis, then Col A is isomorphic to
3
but is not strictly equal to
3.
Yes, because there are
three
pivot columns in A. These columns form a basis in
three
dimensions. Any
3-dimensional
basis spans
3.
Yes, because the column space of a
3×6
matrix is a subspace of
3.
There is a pivot in each row, so the column space is
3-dimensional.
Since any
3-dimensional
subspace of
3
is
3,
Col
A=3.
Is Nul
A=3?
Explain your answer. Choose the correct answer and reasoning below.
No, because although the null space is
3-dimensional,
its basis consists of
three
vectors and not
three.
Therefore, it cannot be equal to
3.
Yes, because a
3×6
matrix exists in
3.
Therefore, if its null space is
3-dimensional
and contained within
3,
it must be equal to
3.
Yes, because the linearly dependent vectors in A form a basis in
three
dimensions. Any basis in
three
dimensions is also a basis for
3.
Therefore, Nul
A=3.
No, because the null space of a
3×6
matrix is a subspace of
6.
Although
dim Nul A=3,
it is not strictly equal to
3
because each vector in Nul A has
six
components. Each vector in
3
has
three
components. Therefore, Nul A is isomorphic to
3,
but not equal.
2.9.17
Determine whether the statement below is true or false. Justify the answer.
If
B=v1,...,vp
is a basis for a subspace H and if
x=c1v1+...+cpvp,
then
c1,...,cp
are the coordinates of x relative to the basis
B.
Choose the correct answer below.
The statement is false because the coordinate vector
[x]B
is composed of the coordinates
c1,...,cp
only if the vector x in
p
is equal to
c1v1+...+cpvp.
The statement is false because x is
v1
coordinates in the direction of
c1,
v2
coordinates in the direction of
c2,
and so on.
The statement is true because the coordinates
c1,...,cp
are the same as the coordinates of x relative to the xy-plane.
The statement is true because any coordinate in a subspace H, with basis
B,
can only be written in one way as a linear combination of basis vectors. The linear combination gives a unique coordinate vector
[x]B
that is composed of the coordinates of x relative to
B.
2.9.17
Determine whether the statement below is true or false. Justify the answer.
If
B=v1,...,vp
is a basis for a subspace H and if
x=c1v1+...+cpvp,
then
c1,...,cp
are the coordinates of x relative to the basis
B.
Choose the correct answer below.
The statement is false because x is
v1
coordinates in the direction of
c1,
v2
coordinates in the direction of
c2,
and so on.
The statement is true because any coordinate in a subspace H, with basis
B,
can only be written in one way as a linear combination of basis vectors. The linear combination gives a unique coordinate vector
[x]B
that is composed of the coordinates of x relative to
B.
The statement is false because the coordinate vector
[x]B
is composed of the coordinates
c1,...,cp
only if the vector x in
p
is equal to
c1v1+...+cpvp.
The statement is true because the coordinates
c1,...,cp
are the same as the coordinates of x relative to the xy-plane.
2.9.17
Determine whether the statement below is true or false. Justify the answer.
If
B=v1,...,vp
is a basis for a subspace H and if
x=c1v1+...+cpvp,
then
c1,...,cp
are the coordinates of x relative to the basis
B.
Choose the correct answer below.
The statement is false because the coordinate vector
[x]B
is composed of the coordinates
c1,...,cp
only if the vector x in
p
is equal to
c1v1+...+cpvp.
The statement is false because x is
v1
coordinates in the direction of
c1,
v2
coordinates in the direction of
c2,
and so on.
The statement is true because the coordinates
c1,...,cp
are the same as the coordinates of x relative to the xy-plane.
The statement is true because any coordinate in a subspace H, with basis
B,
can only be written in one way as a linear combination of basis vectors. The linear combination gives a unique coordinate vector
[x]B
that is composed of the coordinates of x relative to
B.
2.9.18
Determine whether the statement below is true or false. Justify the answer.
If
B
is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in
B.
Choose the correct answer below.
The statement is true. Suppose
B=v1,...,vp
and x is a vector in H that can be generated two ways. Say,
x=c1v1+...+cpvp
and
x=d1v1+...+dpvp,
then
0=xx=c1d1v1+...+cpdpvp.
Therefore,
cp=dp
and x can only be generated in one way.
The statement is true. All bases for a subspace H are linearly independent and therefore each vector in H can only be generated as one unique linear combination of the vectors in
B.
The statement is false. Suppose
B=v1,...,vp
and x is a vector in H. The vector x can be generated in a multiple of ways based on the values of the vectors in the set
B=v1,...,vp.
The statement is false. Bases for a subspace H may be linear dependent and therefore there can be multiple solutions for the same vector x in H.
2.9.18
Determine whether the statement below is true or false. Justify the answer.
If
B
is a basis for a subspace H, then each vector in H can be written in only one way as a linear combination of the vectors in
B.
Choose the correct answer below.
The statement is true. All bases for a subspace H are linearly independent and therefore each vector in H can only be generated as one unique linear combination of the vectors in
B.
The statement is true. Suppose
B=v1,...,vp
and x is a vector in H that can be generated two ways. Say,
x=c1v1+...+cpvp
and
x=d1v1+...+dpvp,
then
0=xx=c1d1v1+...+cpdpvp.
Therefore,
cp=dp
and x can only be generated in one way.
The statement is false. Bases for a subspace H may be linear dependent and therefore there can be multiple solutions for the same vector x in H.
The statement is false. Suppose
B=v1,...,vp
and x is a vector in H. The vector x can be generated in a multiple of ways based on the values of the vectors in the set
B=v1,...,vp.
2.9.19
Determine whether the statement below is true or false. Justify the answer.
Each line in
n
is a one-dimensional subspace of
n.
Choose the correct answer below.
The statement is false. Any subspace of
n
must be at least n-dimensional.
The statement is false. Any subspace of
n
must contain the zero-vector. Therefore, a line can only be a one-dimensional subspace of
n
if it passes through the origin.
The statement is true. Any line in
n
satisfies all three requirements of a subspace.
The statement is true. Any one-dimensional subspace of
n
must be a line.
2.9.19
Determine whether the statement below is true or false. Justify the answer.
Each line in
n
is a one-dimensional subspace of
n.
Choose the correct answer below.
The statement is true. Any one-dimensional subspace of
n
must be a line.
The statement is false. Any subspace of
n
must contain the zero-vector. Therefore, a line can only be a one-dimensional subspace of
n
if it passes through the origin.
The statement is true. Any line in
n
satisfies all three requirements of a subspace.
The statement is false. Any subspace of
n
must be at least n-dimensional.
2.9.21
Determine whether the statement below is true or false. Justify the answer. Here, A is an
m×n
matrix.
The dimension of Col A is the number of pivot columns in A.
Choose the correct answer below.
The statement is false. The dimension of Col A cannot be determined without the size of matrix A.
The statement is false. The number of pivot columns determines the dimension of the null space, not the column space.
The statement is true. The pivot columns of A form a basis for Col A. Therefore, the number of pivot columns of A is the same as the dimension of Col A.
The statement is true. The number of pivot columns is equal to the number of free variables in the equation
Ax=0.
The number of free variables in
Ax=0
is equal to the dimension of the column space.
2.9.21
Determine whether the statement below is true or false. Justify the answer. Here, A is an
m×n
matrix.
The dimension of Col A is the number of pivot columns in A.
Choose the correct answer below.
The statement is false. The number of pivot columns determines the dimension of the null space, not the column space.
The statement is true. The pivot columns of A form a basis for Col A. Therefore, the number of pivot columns of A is the same as the dimension of Col A.
The statement is false. The dimension of Col A cannot be determined without the size of matrix A.
The statement is true. The number of pivot columns is equal to the number of free variables in the equation
Ax=0.
The number of free variables in
Ax=0
is equal to the dimension of the column space.
2.9.22
Determine whether the statement below is true or false. Justify the answer. Here, A is an
m×n
matrix.
The dimension of Nul A is the number of variables in the equation
Ax=0.
Choose the correct answer below.
The statement is true. The dimension of Nul A is the same as the amount of vectors in the set
x=x1,...,xp
that satisfy the equation
Ax=0.
The statement is true. The number of total variables involved in solving the equation
Ax=0
is the dimension of Nul A.
The statement is false. The dimension of Nul A is the number of free variables in the equation
Ax=0.
The statement is false. The dimension of Nul A is the number of variables in the equation
Ax=0
minus the number of free variables in the equation
Ax=0.
2.9.22
Determine whether the statement below is true or false. Justify the answer. Here, A is an
m×n
matrix.
The dimension of Nul A is the number of variables in the equation
Ax=0.
Choose the correct answer below.
The statement is true. The dimension of Nul A is the same as the amount of vectors in the set
x=x1,...,xp
that satisfy the equation
Ax=0.
The statement is false. The dimension of Nul A is the number of free variables in the equation
Ax=0.
The statement is true. The number of total variables involved in solving the equation
Ax=0
is the dimension of Nul A.
The statement is false. The dimension of Nul A is the number of variables in the equation
Ax=0
minus the number of free variables in the equation
Ax=0.
2.9.23
Determine whether the statement below is true or false. Justify the answer. Here, A is an
m×n
matrix.
The dimensions of Col A and Nul A add up to the total number of columns in A.
Choose the correct answer below.
The statement is true. The Rank Theorem states that if matrix A has n columns, then
rank A+dim Nul A=n.
Since rank A is the same as
dim Col A,
the
dimensions of
Col A
and
Nul A
add up to the total number of columns in A.
The statement is true. Col A and Nul A are both subspaces of
n,
where n is the number of columns in matrix A.
The statement is false. The column space and null space sometimes intersect. Therefore, the sum of
dim Col A
and
dim Nul A
is the number of columns of A, only if the
Col A
and
Nul A
are disjoint.
The statement is false. The sum of
dim Col A
and
dim Nul A
is the number of rows of A.
2.9.23
Determine whether the statement below is true or false. Justify the answer. Here, A is an
m×n
matrix.
The dimensions of Col A and Nul A add up to the total number of columns in A.
Choose the correct answer below.
The statement is true. The Rank Theorem states that if matrix A has n columns, then
rank A+dim Nul A=n.
Since rank A is the same as
dim Col A,
the
dimensions of
Col A
and
Nul A
add up to the total number of columns in A.
The statement is false. The column space and null space sometimes intersect. Therefore, the sum of
dim Col A
and
dim Nul A
is the number of columns of A, only if the
Col A
and
Nul A
are disjoint.
The statement is true. Col A and Nul A are both subspaces of
n,
where n is the number of columns in matrix A.
The statement is false. The sum of
dim Col A
and
dim Nul A
is the number of rows of A.
2.9.24
Determine whether the statement below is true or false. Justify the answer. Here, A is an
m×n
matrix.
The dimension of the column space of A is rank A.
Choose the correct answer below.
The statement is true. The rank of matrix A is the dimension of the column space of A.
The statement is true. The rank of matrix A is equal to n, which is also equal to the dimension of the column space of A.
The statement is false. The rank of matrix A is equal to the dimension of the column space of A plus the dimension of the null space of A.
The statement is false. The rank of matrix A is equal to the number of columns plus the number of rows of A, or rank
A=m+n.
2.9.25
Determine whether the statement below is true or false. Justify the answer.
If a set of p vectors spans a p-dimensional subspace H of
n,
then these vectors form a basis of H.
Choose the correct answer below.
The statement is false. Although the set of vectors spans H, there is not enough information to conclude that they form a basis of H.
The statement is false. Only vectors that span H and are linearly independent will form a basis of H. Since the set contains too many vectors, the spanning set cannot possibly be linearly independent.
The statement is true. If a set of p vectors spans a p-dimensional subspace H of
n,
then these vectors must be linearly independent. Any linearly independent spanning set of p vectors forms a basis in p dimensions.
The statement is true. Any spanning set in H will form a basis of H.
2.9.27
If the subspace of all solutions of
Ax=0
has a basis consisting of
three
vectors and if A is a
5×7
matrix, what is the rank of A?
rank
A=4
(Type a whole number.)
2.9.29
If the rank of a
6×8
matrix A is
2,
what is the dimension of the solution space
Ax=0?
The dimension of the solution space is
6.
2.9.29
If the rank of a
5×8
matrix A is
2,
what is the dimension of the solution space
Ax=0?
The dimension of the solution space is
6.
2.9.37
Let
H=Span
v1,v2
and
B=v1,v2.
Show that x is in H, and find the
B-coordinate
vector of
x,
when
v1,
v2,
and x are as below.
v1=
12
5
11
9
,
v2=
15
8
14
12
,
x=
20
13
19
17
Reduce the augmented matrix
v1v2x
to reduced echelon form.
121520
5813
111419
91217
~
1053
0183
000
000
How can it be shown that x is in H?
The augmented matrix is upper triangular and row equivalent to
Bx
,
therefore x is in H because H is the
Spanv1,v2
and
B=v1,v2.
The augmented matrix shows that the system of equations is consistent and therefore x is in H.
The last two rows of the augmented matrix has zero for all entries and this implies that x must be in H.
The first two columns of the augmented matrix are pivot columns and therefore x is in H.
This implies that the
B-coordinate
vector is
x
B
=
53
83
.
2.9.37
Let
H=Span
v1,v2
and
B=v1,v2.
Show that x is in H, and find the
B-coordinate
vector of
x,
when
v1,
v2,
and x are as below.
v1=
13
5
12
7
,
v2=
16
8
15
10
,
x=
21
13
20
15
Reduce the augmented matrix
v1v2x
to reduced echelon form.
131621
5813
121520
71015
~
1053
0183
000
000
How can it be shown that x is in H?
The augmented matrix is upper triangular and row equivalent to
Bx
,
therefore x is in H because H is the
Spanv1,v2
and
B=v1,v2.
The last two rows of the augmented matrix has zero for all entries and this implies that x must be in H.
The first two columns of the augmented matrix are pivot columns and therefore x is in H.
The augmented matrix shows that the system of equations is consistent and therefore x is in H.
This implies that the
B-coordinate
vector is
x
B
=
53
83
.
2.9.37
Let
H=Span
v1,v2
and
B=v1,v2.
Show that x is in H, and find the
B-coordinate
vector of
x,
when
v1,
v2,
and x are as below.
v1=
10
7
9
8
,
v2=
13
10
12
11
,
x=
17
14
16
15
Reduce the augmented matrix
v1v2x
to reduced echelon form.
101317
71014
91216
81115
~
1043
0173
000
000
How can it be shown that x is in H?
The first two columns of the augmented matrix are pivot columns and therefore x is in H.
The augmented matrix shows that the system of equations is consistent and therefore x is in H.
The augmented matrix is upper triangular and row equivalent to
Bx
,
therefore x is in H because H is the
Spanv1,v2
and
B=v1,v2.
The last two rows of the augmented matrix has zero for all entries and this implies that x must be in H.
This implies that the
B-coordinate
vector is
x
B
=
43
73
.
2.9.38
Let
H=Spanv1, v2, v3
and
B=v1, v2, v3.
Show that
B
is a basis for H and x is in H, and find the
B-coordinate
vector of x for the given vectors.
v1=
7
2
9
6
,
v2=
8
3
6
3
,
v3=
9
4
7
3
,
x=
27
7
43
24
Reduce the augmented matrix
v1v2v3x
to reduced echelon form.
78927
2347
96743
63324
~
1006
0103
0011
0000
How can it be shown that
B
is a basis for H?
H is the
Spanv1, v2, v3
and
B=v1, v2, v3
so therefore
B
must form a basis for H.
The first three columns of the augmented matrix are pivot columns and therefore
B
forms a basis for H.
The augmented matrix shows that the system of equations is consistent and therefore
B
forms a basis for H.
The augmented matrix is upper triangular and row equivalent to
Bx
therefore,
B
forms a basis for H.
How can it be shown that x is in H?
The first three columns of the augmented matrix are pivot columns and therefore x is in H.
The augmented matrix is upper triangular and row equivalent to
Bx
,
therefore x is in H because H is the
Spanv1, v2, v3
and
B=v1, v2, v3.
The augmented matrix shows that the system of equations is consistent and therefore x is in H.
The last row of the augmented matrix has zero for all entries and this implies that x must be in H.
The
B-coordinate
vector of is
x
B
=
6
3
1
.
3.1.1
Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.
304
233
041
Compute the determinant using a cofactor expansion across the first row. Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
Using this expansion, the determinant is
(0)(2)+(3)(3)(4)(1)=.
Using this expansion, the determinant is
(0)(2)(3)(3)+(4)(1)=.
Using this expansion, the determinant is
(3)(15)+(0)(2)(4)(8)=.
Using this expansion, the determinant is
(3)(15)(0)(2)+(4)(8)=13.
Compute the determinant using a cofactor expansion down the second column. Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
Using this expansion, the determinant is
(3)(15)+(0)(2)(4)(8)=.
Using this expansion, the determinant is
(3)(15)(0)(2)+(4)(8)=.
Using this expansion, the determinant is
(0)(2)(3)(3)+(4)(1)=.
Using this expansion, the determinant is
(0)(2)+(3)(3)(4)(1)=13.
3.1.1
Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.
304
242
042
Compute the determinant using a cofactor expansion across the first row. Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
Using this expansion, the determinant is
(0)(4)+(4)(6)(4)(2)=.
Using this expansion, the determinant is
(3)(16)(0)(4)+(4)(8)=16.
Using this expansion, the determinant is
(3)(16)+(0)(4)(4)(8)=.
Using this expansion, the determinant is
(0)(4)(4)(6)+(4)(2)=.
Compute the determinant using a cofactor expansion down the second column. Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
Using this expansion, the determinant is
(0)(4)(4)(6)+(4)(2)=.
Using this expansion, the determinant is
(0)(4)+(4)(6)(4)(2)=16.
Using this expansion, the determinant is
(3)(16)+(0)(4)(4)(8)=.
Using this expansion, the determinant is
(3)(16)(0)(4)+(4)(8)=.
3.1.1
Compute the determinant using a cofactor expansion across the first row. Also compute the determinant by a cofactor expansion down the second column.
204
343
052
Compute the determinant using a cofactor expansion across the first row. Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
Using this expansion, the determinant is
(0)(6)(4)(4)+(5)(6)=.
Using this expansion, the determinant is
(2)(23)(0)(6)+(4)(15)=14.
Using this expansion, the determinant is
(2)(23)+(0)(6)(4)(15)=.
Using this expansion, the determinant is
(0)(6)+(4)(4)(5)(6)=.
Compute the determinant using a cofactor expansion down the second column. Select the correct choice below and fill in the answer box to complete your choice.
(Simplify your answer.)
Using this expansion, the determinant is
(2)(23)+(0)(6)(4)(15)=.
Using this expansion, the determinant is
(0)(6)+(4)(4)(5)(6)=14.
Using this expansion, the determinant is
(0)(6)(4)(4)+(5)(6)=.
Using this expansion, the determinant is
(2)(23)(0)(6)+(4)(15)=.
3.1.9
Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation.
9005
8733
3000
7311
9005
8733
3000
7311
=30
(Simplify your answer.)
3.1.9
Compute the determinant by cofactor expansion. At each step, choose a row or column that involves the least amount of computation.
5005
6732
2000
3311
5005
6732
2000
3311
=20
(Simplify your answer.)
3.1.10
Compute the following determinant by cofactor expansions. At each step, choose the row or column that involves the least amount of computation.
3335
0040
1645
5053
The determinant is
120.
3.1.10
Compute the following determinant by cofactor expansions. At each step, choose the row or column that involves the least amount of computation.
4535
0020
3275
6053
The determinant is
138.
3.1.15
The expansion of a
3×3
determinant can be remembered by this device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals. Add the downward diagonal products and subtract the upward products. Use this method to compute the following determinant.
142
501
032
+
+
+
a11
a12
a13
a11
a12
a21
a22
a23
a21
a22
a31
a32
a33
a31
a32
142
501
032
=13
3.1.15
The expansion of a
3×3
determinant can be remembered by this device. Write a second copy of the first two columns to the right of the matrix, and compute the determinant by multiplying entries on six diagonals. Add the downward diagonal products and subtract the upward products. Use this method to compute the following determinant.
542
340
043
+
+
+
a11
a12
a13
a11
a12
a21
a22
a23
a21
a22
a31
a32
a33
a31
a32
542
340
043
=48
3.1.20
State the row operation performed below and describe how it affects the determinant.
ab
cd
,
ab
3c3d
What row operation was performed?
The row operation
scales row 2 by one-third.
The row operation
adds 3 to row 2.
The row operation
subtracts 3 from row 2.
The row operation
scales row 2 by 3.
How does this affect the determinant?
The determinant is divided by
3.
The determinant is multiplied by 3.
The determinant is 0.
The determinant is unchanged.
3.1.20
State the row operation performed below and describe how it affects the determinant.
ab
cd
,
cd
ab
What row operation was performed?
The row operation
multiplies row 1 by d and row 2 by b.
The row operation
multiplies row 1 by c and row 2 by a.
The row operation
scales row 1 by c and row 2 by a.
The row operation
swaps rows 1 and 2.
How does this affect the determinant?
The determinant is 0.
The sign of the determinant is reversed.
The determinant is divided by
2.
The determinant is unchanged.
3.1.21
Explore the effects of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
84
25
,
84
2+8k5+4k
What is the elementary row operation?
Replace row 2 with k times row 2.
Replace row 2 with k times row 1 plus row 2.
Replace row 2 with k times row 1.
Replace row 2 with row 1 plus k times row 2.
How does the row operation affect the determinant?
The determinant is decreased by
32k.
The determinant is increased by
64k.
The determinant is increased by
32k.
The determinant does not change.
3.1.21
Explore the effects of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
26
92
,
26
9+2k2+6k
What is the elementary row operation?
Replace row 2 with k times row 1 plus row 2.
Replace row 2 with row 1 plus k times row 2.
Replace row 2 with k times row 2.
Replace row 2 with k times row 1.
How does the row operation affect the determinant?
The determinant is increased by
24k.
The determinant is increased by
12k.
The determinant is decreased by
12k.
The determinant does not change.
3.1.21
Explore the effects of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
36
62
,
36
6+3k2+6k
What is the elementary row operation?
Replace row 2 with k times row 2.
Replace row 2 with k times row 1 plus row 2.
Replace row 2 with row 1 plus k times row 2.
Replace row 2 with k times row 1.
How does the row operation affect the determinant?
The determinant is decreased by
18k.
The determinant is increased by
18k.
The determinant is increased by
36k.
The determinant does not change.
3.1.22
Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
ab
cd
,
a+kcb+kd
cd
What is the elementary row operation?
Row 2 is replaced with the sum of itself and k times row 1.
Row 1 is replaced with the sum of itself and k times row 2.
Row 2 is multiplied by k.
Rows 1 and 2 are interchanged.
Row 1 is multiplied by k.
How does the row operation affect the determinant?
It multiplies the determinant by k.
It changes the sign of the determinant.
It increases the determinant by k.
It does not change the determinant.
3.1.22
Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
ab
cd
,
a+kcb+kd
cd
What is the elementary row operation?
Rows 1 and 2 are interchanged.
Row 1 is replaced with the sum of itself and k times row 2.
Row 2 is replaced with the sum of itself and k times row 1.
Row 1 is multiplied by k.
Row 2 is multiplied by k.
How does the row operation affect the determinant?
It increases the determinant by k.
It changes the sign of the determinant.
It multiplies the determinant by k.
It does not change the determinant.
3.1.23
Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
365
978
abc
,
abc
978
365
What is the elementary row operation?
Row
1
is replaced with the sum of rows
1
and
2.
Row
1
is replaced with the sum of rows
1
and
3.
Row
3
is replaced with the sum of rows
2
and
3.
Row
3
is replaced with the sum of rows
1
and
3.
Rows
1
and
3
are interchanged.
Rows
1
and
2
are interchanged.
Rows
2
and
3
are interchanged.
How does the row operation affect the determinant?
It changes the sign of the determinant.
It increases the determinant by 1.
It multiplies the determinant by 2.
It does not change the determinant.
3.1.23
Explore the effect of an elementary row operation on the determinant of a matrix. State the row operation and describe how it affects the determinant.
471
abc
265
,
abc
471
265
What is the elementary row operation?
Row
1
is replaced with the sum of rows
1
and
2.
Rows
2
and
3
are interchanged.
Row
2
is replaced with the sum of rows
1
and
2.
Row
1
is replaced with the sum of rows
1
and
3.
Rows
1
and
2
are interchanged.
Row
2
is replaced with the sum of rows
2
and
3.
Rows
1
and
3
are interchanged.
How does the row operation affect the determinant?
It changes the sign of the determinant.
It multiplies the determinant by 2.
It increases the determinant by 1.
It does not change the determinant.
3.1.26
Compute the determinant of the following elementary matrix.
010
100
001
010
100
001
=1
(Simplify your answer.)
3.1.37
Let
A=
12
79
.
Write
4A.
Is
det(4A)
equal to
4det(A)?
4A=
48
2836
(Type an integer or decimal for each matrix element.)
Select the correct choice below and fill in the answer box(es) to complete your choice.
Yes,
det(4A)
is equal to
4det(A).
The value of both expressions is
.
No,
det(4A)
is not equal to
4det(A).
The value of
det(4A)
is
80,
whereas the value of
4det(A)
is
20.
3.1.37
Let
A=
46
52
.
Write
4A.
Is
det(4A)
equal to
4det(A)?
4A=
1624
208
(Type an integer or decimal for each matrix element.)
Select the correct choice below and fill in the answer box(es) to complete your choice.
No,
det(4A)
is not equal to
4det(A).
The value of
det(4A)
is
352,
whereas the value of
4det(A)
is
88.
Yes,
det(4A)
is equal to
4det(A).
The value of both expressions is
.
3.1.37
Let
A=
51
89
.
Write
4A.
Is
det(4A)
equal to
4det(A)?
4A=
204
3236
(Type an integer or decimal for each matrix element.)
Select the correct choice below and fill in the answer box(es) to complete your choice.
Yes,
det(4A)
is equal to
4det(A).
The value of both expressions is
.
No,
det(4A)
is not equal to
4det(A).
The value of
det(4A)
is
592,
whereas the value of
4det(A)
is
148.
3.1.39
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
An
n×n
determinant is defined by determinants of
(n1)×(n1)
submatrices.
Choose the correct answer below.
The statement is true. The determinant of an
n×n
matrix A can be computed by a cofactor expansion across any row or down any column. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size
(n1)×(n1).
The statement is false. Although determinants of
(n1)×(n1)
submatrices can be used to find
n×n
determinants, they are not involved in the definition of
n×n
determinants.
The statement is false. An
n×n
determinant is defined by determinants of
(n1)×(n1)
submatrices only when
n>3.
Determinants of
1×1,
2×2,
and
3×3
matrices are defined separately.
The statement is true. The determinant of an
n×n
matrix A can be computed by a cofactor expansion along either diagonal. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size
(n1)×(n1).
3.1.39
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
An
n×n
determinant is defined by determinants of
(n1)×(n1)
submatrices.
Choose the correct answer below.
The statement is false. An
n×n
determinant is defined by determinants of
(n1)×(n1)
submatrices only when
n>3.
Determinants of
1×1,
2×2,
and
3×3
matrices are defined separately.
The statement is true. The determinant of an
n×n
matrix A can be computed by a cofactor expansion along either diagonal. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size
(n1)×(n1).
The statement is false. Although determinants of
(n1)×(n1)
submatrices can be used to find
n×n
determinants, they are not involved in the definition of
n×n
determinants.
The statement is true. The determinant of an
n×n
matrix A can be computed by a cofactor expansion across any row or down any column. Each term in any such expansion includes a cofactor that involves the determinant of a submatrix of size
(n1)×(n1).
3.1.40
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The (i,j)-cofactor of a matrix A is the matrix
Aij
obtained by deleting from A its ith row and jth column.
Choose the correct answer below.
The statement is true. It is the definition of the (i,j)-cofactor of a matrix A.
The statement is false. The (i,j)-cofactor of A is the number
Cij=detAij,
where
Aij
is the submatrix obtained by deleting from A its ith row and jth column.
The statement is false. The (i,j)-cofactor of a matrix A is the matrix
Aij
obtained by deleting from A its jth row and ith column.
The statement is false. The (i,j)-cofactor of A is the number
Cij=(1)i+jdetAij,
where
Aij
is the submatrix obtained by deleting from A its ith row and jth column.
3.1.40
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The (i,j)-cofactor of a matrix A is the matrix
Aij
obtained by deleting from A its ith row and jth column.
Choose the correct answer below.
The statement is false. The (i,j)-cofactor of A is the number
Cij=(1)i+jdetAij,
where
Aij
is the submatrix obtained by deleting from A its ith row and jth column.
The statement is false. The (i,j)-cofactor of A is the number
Cij=detAij,
where
Aij
is the submatrix obtained by deleting from A its ith row and jth column.
The statement is false. The (i,j)-cofactor of a matrix A is the matrix
Aij
obtained by deleting from A its jth row and ith column.
The statement is true. It is the definition of the (i,j)-cofactor of a matrix A.
3.1.40
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The (i,j)-cofactor of a matrix A is the matrix
Aij
obtained by deleting from A its ith row and jth column.
Choose the correct answer below.
The statement is false. The (i,j)-cofactor of a matrix A is the matrix
Aij
obtained by deleting from A its jth row and ith column.
The statement is false. The (i,j)-cofactor of A is the number
Cij=(1)i+jdetAij,
where
Aij
is the submatrix obtained by deleting from A its ith row and jth column.
The statement is true. It is the definition of the (i,j)-cofactor of a matrix A.
The statement is false. The (i,j)-cofactor of A is the number
Cij=detAij,
where
Aij
is the submatrix obtained by deleting from A its ith row and jth column.
3.1.42
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
Choose the correct answer below.
The statement is true. Cofactor expansion along the row (or column) with the most zeros of a triangular matrix produces a determinant equal to the sum of the entries along the main diagonal.
The statement is false. The determinant of a triangular matrix is the product of the entries along the main diagonal.
The statement is false. The determinant of a matrix is the arithmetic mean of the entries along the main diagonal.
The statement is true. The determinant of A is the following finite series.
det A=j=1n(1)1+ja1jdet A1j
In a triangular matrix, this series simplifies to the sum of the entries along the main diagonal.
3.1.42
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
Choose the correct answer below.
The statement is false. The determinant of a triangular matrix is the product of the entries along the main diagonal.
The statement is true. The determinant of A is the following finite series.
det A=j=1n(1)1+ja1jdet A1j
In a triangular matrix, this series simplifies to the sum of the entries along the main diagonal.
The statement is true. Cofactor expansion along the row (or column) with the most zeros of a triangular matrix produces a determinant equal to the sum of the entries along the main diagonal.
The statement is false. The determinant of a matrix is the arithmetic mean of the entries along the main diagonal.
3.1.42
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The determinant of a triangular matrix is the sum of the entries on the main diagonal.
Choose the correct answer below.
The statement is false. The determinant of a matrix is the arithmetic mean of the entries along the main diagonal.
The statement is true. The determinant of A is the following finite series.
det A=j=1n(1)1+ja1jdet A1j
In a triangular matrix, this series simplifies to the sum of the entries along the main diagonal.
The statement is false. The determinant of a triangular matrix is the product of the entries along the main diagonal.
The statement is true. Cofactor expansion along the row (or column) with the most zeros of a triangular matrix produces a determinant equal to the sum of the entries along the main diagonal.
3.2.1
State which property of determinants is illustrated in this equation.
602
1826
579
=
1826
602
579
Choose the correct answer below.
If one row of A is multiplied by k to produce B, then
det B=kdet A.
If A and B are square matrices, then det AB=(det A)(det B).
If two rows of A are interchanged to produce B, then det B=det A.
If a multiple of one row of A is added to another row to produce matrix B, then det B=det A.
3.2.1
State which property of determinants is illustrated in this equation.
249
426
501
=
426
249
501
Choose the correct answer below.
If two rows of A are interchanged to produce B, then det B=det A.
If A and B are square matrices, then det AB=(det A)(det B).
If one row of A is multiplied by k to produce B, then
det B=kdet A.
If a multiple of one row of A is added to another row to produce matrix B, then det B=det A.
3.2.2
The equation below illustrates a property of determinants. State the property.
369
355
133
=3
123
355
133
Multiplying a row by 3 divides the determinant by 3.
Multiplying a row by 3 multiplies the determinant by 3.
Factoring 3 out of the entire matrix divides the determinant by 3.
Dividing a row by 3 multiplies the determinant by 3.
3.2.2
The equation below illustrates a property of determinants. State the property.
369
355
133
=3
123
355
133
Multiplying a row by 3 multiplies the determinant by 3.
Multiplying a row by 3 divides the determinant by 3.
Dividing a row by 3 multiplies the determinant by 3.
Factoring 3 out of the entire matrix divides the determinant by 3.
3.2.4
State which property of determinants is illustrated in this equation.
443
1298
451
=
443
0211
451
Choose the correct answer below.
If two rows of A are interchanged to produce B, then
det B=det A.
If A and B are square matrices, then
det AB=(det A)(det B).
If a multiple of one row of A is added to another row to produce matrix B, then
det B=det A.
If one row of A is multiplied by k to produce B, then
det B=kdet A.
3.2.4
State which property of determinants is illustrated in this equation.
852
2445
027
=
852
01911
027
Choose the correct answer below.
If two rows of A are interchanged to produce B, then
det B=det A.
If a multiple of one row of A is added to another row to produce matrix B, then
det B=det A.
If one row of A is multiplied by k to produce B, then
det B=kdet A.
If A and B are square matrices, then
det AB=(det A)(det B).
3.2.5
Find the determinant by row reduction to echelon form.
157
165
147
Use row operations to reduce the matrix to echelon form.
157
165
147
~
100
010
001
Find the determinant of the given matrix.
157
165
147
=12
(Simplify your answer.)
3.2.5
Find the determinant by row reduction to echelon form.
157
145
287
Use row operations to reduce the matrix to echelon form.
157
145
287
~
100
010
001
Find the determinant of the given matrix.
157
145
287
=3
(Simplify your answer.)
3.2.5
Find the determinant by row reduction to echelon form.
157
165
147
Use row operations to reduce the matrix to echelon form.
157
165
147
~
100
010
001
Find the determinant of the given matrix.
157
165
147
=12
(Simplify your answer.)
3.2.7
Find the determinant by row reduction to echelon form.
1130
4332
3142
1382
Use row operations to reduce the matrix to echelon form.
1130
4332
3142
1382
~
1002625
01045
001225
0000
Find the determinant of the given matrix.
1130
4332
3142
1382
=0
(Simplify your answer.)
3.2.15
Find the determinant below, where
abc
def
ghi
=6.
abc
def
4g4h4i
abc
def
4g4h4i
=24
(Simplify your answer.)
3.2.15
Find the determinant below, where
abc
def
ghi
=8.
abc
def
5g5h5i
abc
def
5g5h5i
=40
(Simplify your answer.)
3.2.20
If
abc
def
ghi
=20,
find the determinant of
abc
ghi
def
.
The determinant is
20.
3.2.20
If
abc
def
ghi
=9,
find the determinant of
abc
ghi
def
.
The determinant is
9.
3.2.20
If
abc
def
ghi
=4,
find the determinant of
abc
ghi
def
.
The determinant is
4.
3.2.21
Use determinants to find out if the matrix is invertible.
501
132
053
The determinant of the matrix is
0.
(Simplify your answer.)
Is the matrix invertible? Choose the correct answer below.
The matrix is not invertible.
The matrix is invertible.
3.2.22
Use determinants to find out if the matrix is invertible.
2051
4152
0253
The determinant of the matrix is
60.
(Simplify your answer.)
Is the matrix invertible?
The matrix is not invertible because the determinant is not zero.
The matrix is invertible because the determinant of the matrix is not zero.
The matrix is invertible because the determinant is defined.
The matrix is not invertible because the determinant is defined.
3.2.22
Use determinants to find out if the matrix is invertible.
1032
294
0156
The determinant of the matrix is
36.
(Simplify your answer.)
Is the matrix invertible?
The matrix is invertible because the determinant is defined.
The matrix is not invertible because the determinant is not zero.
The matrix is not invertible because the determinant is defined.
The matrix is invertible because the determinant of the matrix is not zero.
3.2.22
Use determinants to find out if the matrix is invertible.
2013
436
059
The determinant of the matrix is
36.
(Simplify your answer.)
Is the matrix invertible?
The matrix is not invertible because the determinant is defined.
The matrix is invertible because the determinant of the matrix is not zero.
The matrix is invertible because the determinant is defined.
The matrix is not invertible because the determinant is not zero.
3.2.25
Use determinants to decide if the set of vectors is linearly independent.
3
3
1
,
2
4
3
,
3
0
4
The determinant of the matrix whose columns are the given vectors is
9.
(Simplify your answer.)
Is the set of vectors linearly independent?
The set of vectors is linearly dependent, because the determinant exists.
The set of vectors is linearly dependent, because the determinant is not zero.
The set of vectors is linearly independent, because the determinant is not zero.
The set of vectors is linearly independent, because the determinant exists.
3.2.25
Use determinants to decide if the set of vectors is linearly independent.
2
7
3
,
1
6
2
,
2
0
6
The determinant of the matrix whose columns are the given vectors is
22.
(Simplify your answer.)
Is the set of vectors linearly independent?
The set of vectors is linearly independent, because the determinant is not zero.
The set of vectors is linearly dependent, because the determinant exists.
The set of vectors is linearly dependent, because the determinant is not zero.
The set of vectors is linearly independent, because the determinant exists.
3.2.25
Use determinants to decide if the set of vectors is linearly independent.
2
2
6
,
3
1
2
,
2
0
1
The determinant of the matrix whose columns are the given vectors is
24.
(Simplify your answer.)
Is the set of vectors linearly independent?
The set of vectors is linearly independent, because the determinant exists.
The set of vectors is linearly independent, because the determinant is not zero.
The set of vectors is linearly dependent, because the determinant exists.
The set of vectors is linearly dependent, because the determinant is not zero.
3.2.28
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If det A is zero, then two rows or two columns are the same, or a row or a column is zero.
Choose the correct answer below.
The statement is false. If
A=
26
13
,
then det
A=0
and the rows and columns are all distinct and not full of zeros.
The statement is true. If det A is zero, then the columns of A are linearly independent. If one column is zero, or two columns are the same, then the columns are linearly dependent.
The statement is true. If
A=
23
23
and
B=
12
00
,
then det
A=0
and det
B=0.
The statement is false. The determinant depends on the columns of A. It is possible for two rows to be the same and for the determinant to be nonzero.
3.2.28
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If det A is zero, then two rows or two columns are the same, or a row or a column is zero.
Choose the correct answer below.
The statement is false. If
A=
26
13
,
then det
A=0
and the rows and columns are all distinct and not full of zeros.
The statement is false. The determinant depends on the columns of A. It is possible for two rows to be the same and for the determinant to be nonzero.
The statement is true. If det A is zero, then the columns of A are linearly independent. If one column is zero, or two columns are the same, then the columns are linearly dependent.
The statement is true. If
A=
23
23
and
B=
12
00
,
then det
A=0
and det
B=0.
3.2.28
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If det A is zero, then two rows or two columns are the same, or a row or a column is zero.
Choose the correct answer below.
The statement is true. If
A=
23
23
and
B=
12
00
,
then det
A=0
and det
B=0.
The statement is false. If
A=
26
13
,
then det
A=0
and the rows and columns are all distinct and not full of zeros.
The statement is false. The determinant depends on the columns of A. It is possible for two rows to be the same and for the determinant to be nonzero.
The statement is true. If det A is zero, then the columns of A are linearly independent. If one column is zero, or two columns are the same, then the columns are linearly dependent.
3.2.29
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If the columns of A are linearly dependent, then det
A=0.
Choose the correct answer below.
The statement is false. If det
A=0,
then A is invertible.
The statement is false. The columns of I are linearly dependent, yet det I
=1.
The statement is true. If the columns of A are linearly dependent, then one of the columns is equal to another.
The statement is true. If the columns of A are linearly dependent, then A is not invertible.
3.2.29
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If the columns of A are linearly dependent, then det
A=0.
Choose the correct answer below.
The statement is false. If det
A=0,
then A is invertible.
The statement is true. If the columns of A are linearly dependent, then A is not invertible.
The statement is false. The columns of I are linearly dependent, yet det I
=1.
The statement is true. If the columns of A are linearly dependent, then one of the columns is equal to another.
3.2.29
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If the columns of A are linearly dependent, then det
A=0.
Choose the correct answer below.
The statement is true. If the columns of A are linearly dependent, then A is not invertible.
The statement is false. The columns of I are linearly dependent, yet det I
=1.
The statement is true. If the columns of A are linearly dependent, then one of the columns is equal to another.
The statement is false. If det
A=0,
then A is invertible.
3.2.32
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The determinant of A is the product of the pivots in any echelon form U of A, multiplied by
(1)r,
where r is the number of row interchanges made during row reduction from A to U.
Choose the correct answer below.
The statement is false. Reduction to an echelon form may also include scaling a row by a nonzero constant, which can change the value of the determinant.
The statement is false. The determinant is the product of the number of pivots and
(1)r.
The statement is true. If
A=
23
04
,
then det
A=8.
The statement is true. The determinant is the product of the entries on the diagonal and the pivots are all on the diagonal.
3.2.32
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The determinant of A is the product of the pivots in any echelon form U of A, multiplied by
(1)r,
where r is the number of row interchanges made during row reduction from A to U.
Choose the correct answer below.
The statement is true. The determinant is the product of the entries on the diagonal and the pivots are all on the diagonal.
The statement is false. The determinant is the product of the number of pivots and
(1)r.
The statement is false. Reduction to an echelon form may also include scaling a row by a nonzero constant, which can change the value of the determinant.
The statement is true. If
A=
23
04
,
then det
A=8.
3.2.33
Let A and B be
n×n
matrices. Determine whether the statement below is true or false. Justify the answer.
det(A+B)=det
A+det
B
Choose the correct answer below.
The statement is true. If
A=
20
10
and
B=
30
50
,
then
det(A+B)=0
and
det A+det B=0.
The statement is true. Determinants are linear transformations.
The statement is false. If
A=
10
01
and
B=
10
01
,
then
det(A+B)=0
and det
A+det
B=2.
The statement is false.
det(A+B)=(det
A)(det B)
3.2.35
Compute
det B4
where
B=
102
123
231
.
             
det B4=6561
(Simplify your answer.)
3.2.35
Compute
det B4
where
B=
102
212
121
.
             
det B4=81
(Simplify your answer.)
3.2.35
Compute
det B4
where
B=
201
213
121
.
             
det B4=2401
(Simplify your answer.)
3.2.38
Suppose that A is a square matrix such that det
A3=0.
Explain why A cannot be invertible.
Since the desired conclusion is that A is not invertible, it will be helpful to identify any conditions that are logically equivalent to the condition that A is not invertible. Which of the following conditions is logically equivalent to the condition that A is not invertible?
det
AB=(det
A)(det B)
det
A=0
det
A0
det
AT=det
A
What property can be applied to the left side of the equation det
A3=0
in order to yield the desired result?
det
A0
if and only if A is invertible.
det
A3=(det
A)(det A)(det A)
det
AT=det
A
If two rows of A are interchanged to produce B, then det
B=det
A.
What should be done next to establish that A is not invertible? Select the correct choice below and fill in the answer box(es) within your choice.
(Type an integer or a decimal.)
By the zero factor property, det
A=0.
Because det
B=,
det
A=.
By the zero factor property, det
AT=.
Because det
AB=,
det
A=.
Thus, A is not invertible.
3.2.38
Suppose that A is a square matrix such that det
A3=0.
Explain why A cannot be invertible.
Since the desired conclusion is that A is not invertible, it will be helpful to identify any conditions that are logically equivalent to the condition that A is not invertible. Which of the following conditions is logically equivalent to the condition that A is not invertible?
det
A0
det
AB=(det
A)(det B)
det
AT=det
A
det
A=0
What property can be applied to the left side of the equation det
A3=0
in order to yield the desired result?
If two rows of A are interchanged to produce B, then det
B=det
A.
det
AT=det
A
det
A0
if and only if A is invertible.
det
A3=(det
A)(det A)(det A)
What should be done next to establish that A is not invertible? Select the correct choice below and fill in the answer box(es) within your choice.
(Type an integer or a decimal.)
By the zero factor property, det
A=0.
By the zero factor property, det
AT=.
Because det
B=,
det
A=.
Because det
AB=,
det
A=.
Thus, A is not invertible.
3.2.51
Suppose A is an
n×n
matrix and a computer suggests that
det A=6
and
det A1=1.
Should you trust these answers? Why or why not?
Choose the correct answer below.
Yes, you should trust the answers because both determinants are non-zero and
det A1
is the same sign as
det A.
No, you should not trust the answers because
det A×det A1
should always equal 1.
No, you should not trust the answers because
det A×det A1
should always equal
1.
No, you should not trust the answers because
det A
should always be the opposite sign of
det A1.
3.2.51
Suppose A is an
n×n
matrix and a computer suggests that
det A=1
and
det A1=7.
Should you trust these answers? Why or why not?
Choose the correct answer below.
No, you should not trust the answers because
det A
should always be the opposite sign of
det A1.
Yes, you should trust the answers because both determinants are non-zero and
det A1
is the same sign as
det A.
No, you should not trust the answers because
det A×det A1
should always equal
1.
No, you should not trust the answers because
det A×det A1
should always equal 1.
3.3.1
Use Cramer's rule to compute the solutions of the system.
 2x1+3x2=2
 2x1+7x2=26
What is the solution of the system?
x1=8
x2=6
3.3.1
Use Cramer's rule to compute the solutions of the system.
 4x1+3x2=5
 5x1+6x2=5
What is the solution of the system?
x1=5
x2=5
3.3.1
Use Cramer's rule to compute the solutions of the system.
 2x1+3x2=3
 5x1+7x2=5
What is the solution of the system?
x1=6
x2=5
3.3.3
Use Cramer's rule to compute the solutions of the system.
 2x1+7x2=
 6
  4x1+7x2=
 5
What is the solution of the system?
x1=16
x2=1721
(Type integers or simplified fractions.)
3.3.7
Determine the values of the parameter s for which the system has a unique solution, and describe the solution.
4sx1+7x2
=
3
8x1+3sx2
=
3
Choose the correct answer below.
s±143;
x1=3(s2)3s214;
x2=3(3s+7)43s214
s0;
x1=3(3s+7)43s214;
x2=3(s2)3s214
s0;
x1=3(s2)3s214;
x2=3(3s+7)43s214
s±143;
x1=3(3s+7)43s214;
x2=3(s2)3s214
3.3.7
Determine the values of the parameter s for which the system has a unique solution, and describe the solution.
3sx1+4x2
=
5
9x1+4sx2
=
4
Choose the correct answer below.
s0;
x1=5s+43s23;
x2=4s154s23
s0;
x1=4s154s23;
x2=5s+43s23
s±3;
x1=4s154s23;
x2=5s+43s23
s±3;
x1=5s+43s23;
x2=4s154s23
3.3.7
Determine the values of the parameter s for which the system has a unique solution, and describe the solution.
2sx1+4x2
=
5
9x1+3sx2
=
4
Choose the correct answer below.
s0;
x1=8s456s26;
x2=15s+166s26
s±6;
x1=15s+166s26;
x2=8s456s26
s±6;
x1=8s456s26;
x2=15s+166s26
s0;
x1=15s+166s26;
x2=8s456s26
3.3.11
Compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix.
A=
031
300
221
The adjugate of the given matrix is adj A=
010
323
669
.
(Type an integer or simplified fraction for each matrix element.)
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
0130
1231
223
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
3.3.11
Compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix.
A=
021
300
211
The adjugate of the given matrix is adj A=
010
323
346
.
(Type an integer or simplified fraction for each matrix element.)
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
0130
1231
1432
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
3.3.11
Compute the adjugate of the given matrix, and then use the Inverse Formula to give the inverse of the matrix.
A=
031
600
221
The adjugate of the given matrix is adj A=
010
626
12618
.
(Type an integer or simplified fraction for each matrix element.)
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The inverse matrix is
A1=
0160
1131
213
.
(Type an integer or simplified fraction for each matrix element.)
The matrix is not invertible.
3.3.18
Suppose that all the entries in A are integers and det
A=1.
Explain why all the entries in
A1
are integers.
Choose the correct answer below.
Each cofactor in A is an integer because it is just a sum of products of entries of A. Hence all the entries in adj A are integers. Since det
A=1,
the inverse formula shows that all the entries in
A1
are integers.
The inverse of A is always equal to the reciprocal of the determinant multiplied by matrix A. Since det
A=1,
the reciprocal is also equal to one, so the inverse of A is equal to matrix A.
Each cofactor in A is a fraction with the same denominator because it is just a sum of quotient of entries of A. All of the fractions sum to an integer. Since det
A=1,
the inverse formula shows that all the entries in
A1
are integers.
The inverse of A is always equal to the negative reciprocal of the determinant multiplied by matrix A. Since det
A=1,
the reciprocal is also equal to one, so each component of the inverse matrix is equal to the negative of the components of matrix A.
3.3.18
Suppose that all the entries in A are integers and det
A=1.
Explain why all the entries in
A1
are integers.
Choose the correct answer below.
The inverse of A is always equal to the reciprocal of the determinant multiplied by matrix A. Since det
A=1,
the reciprocal is also equal to one, so the inverse of A is equal to matrix A.
Each cofactor in A is an integer because it is just a sum of products of entries of A. Hence all the entries in adj A are integers. Since det
A=1,
the inverse formula shows that all the entries in
A1
are integers.
Each cofactor in A is a fraction with the same denominator because it is just a sum of quotient of entries of A. All of the fractions sum to an integer. Since det
A=1,
the inverse formula shows that all the entries in
A1
are integers.
The inverse of A is always equal to the negative reciprocal of the determinant multiplied by matrix A. Since det
A=1,
the reciprocal is also equal to one, so each component of the inverse matrix is equal to the negative of the components of matrix A.
5.1.2
Is
λ=2
an eigenvalue of
A=
22
40
?
Why or why not?
Choose the correct answer below.
Yes,
λ
is an eigenvalue of A because
(AλI)
is invertible.
Yes,
λ
is an eigenvalue of A because
Ax=λx
has a nontrivial solution.
Yes,
λ
is an eigenvalue of A because
λAx=0
only has the trivial solution.
No,
λ
is not an eigenvalue of A because
Ax=λx
has a nontrivial solution.
No,
λ
is not an eigenvalue of A because
λAx=0
only has the trivial solution.
No,
λ
is not an eigenvalue of A because
Ax=λx
only has the trivial solution.
5.1.2
Is
λ=7
an eigenvalue of
A=
95
22
?
Why or why not?
Choose the correct answer below.
No,
λ
is not an eigenvalue of A because
λAx=0
only has the trivial solution.
No,
λ
is not an eigenvalue of A because
Ax=λx
has a nontrivial solution.
Yes,
λ
is an eigenvalue of A because
λAx=0
only has the trivial solution.
Yes,
λ
is an eigenvalue of A because
(AλI)
is invertible.
Yes,
λ
is an eigenvalue of A because
Ax=λx
has a nontrivial solution.
No,
λ
is not an eigenvalue of A because
Ax=λx
only has the trivial solution.
5.1.4
Is
v=
1
1
an eigenvector of
A=
95
22
?
If so, find the eigenvalue.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
Yes, v is an eigenvector of A. The eigenvalue is
λ=4.
No, v is not an eigenvector of A.
5.1.4
Is
v=
2
1
an eigenvector of
A=
14
33
?
If so, find the eigenvalue.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
Yes, v is an eigenvector of A. The eigenvalue is
λ=3.
No, v is not an eigenvector of A.
5.1.6
Is
v=
0
1
0
an eigenvector of
A=
401
210
201
?
If so, find the eigenvalue.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
Yes, v is an eigenvector of A. The eigenvalue is
λ=1.
No, v is not an eigenvector of A.
5.1.6
Is
v=
1
5
2
an eigenvector of
A=
102
254
022
?
If so, find the eigenvalue.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
Yes, v is an eigenvector of A. The eigenvalue is
λ=3.
No, v is not an eigenvector of A.
5.1.6
Is
v=
3
2
1
an eigenvector of
A=
433
232
102
?
If so, find the eigenvalue.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
Yes, v is an eigenvector of A. The eigenvalue is
λ=5.
No, v is not an eigenvector of A.
5.1.7
Is
λ=8
an eigenvalue of
703
275
343
?
If so, find one corresponding eigenvector.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
Yes,
λ=8
is an eigenvalue of
703
275
343
.
One corresponding eigenvector is
3
1
1
.
(Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element.)
No,
λ=8
is not an eigenvalue of
703
275
343
.
5.1.7
Is
λ=7
an eigenvalue of
603
366
421
?
If so, find one corresponding eigenvector.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
Yes,
λ=7
is an eigenvalue of
603
366
421
.
One corresponding eigenvector is
3
3
1
.
(Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element.)
No,
λ=7
is not an eigenvalue of
603
366
421
.
5.1.7
Is
λ=5
an eigenvalue of
401
243
432
?
If so, find one corresponding eigenvector.
Select the correct choice below and, if necessary, fill in the answer box within your choice.
Yes,
λ=5
is an eigenvalue of
401
243
432
.
One corresponding eigenvector is
1
1
1
.
(Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element.)
No,
λ=5
is not an eigenvalue of
401
243
432
.
5.1.9
Find a basis for the eigenspace corresponding to each listed eigenvalue of A below.
A=
30
14
,
λ=4,3
A basis for the eigenspace corresponding to
λ=4
is
0
1
.
(Use a comma to separate answers as needed.)
A basis for the eigenspace corresponding to
λ=3
is
1
1
.
(Use a comma to separate answers as needed.)
5.1.9
Find a basis for the eigenspace corresponding to each listed eigenvalue of A below.
A=
10
12
,
λ=2,1
A basis for the eigenspace corresponding to
λ=2
is
0
1
.
(Use a comma to separate answers as needed.)
A basis for the eigenspace corresponding to
λ=1
is
1
1
.
(Use a comma to separate answers as needed.)
5.1.9
Find a basis for the eigenspace corresponding to each listed eigenvalue of A below.
A=
10
45
,
λ=5,1
A basis for the eigenspace corresponding to
λ=5
is
0
1
.
(Use a comma to separate answers as needed.)
A basis for the eigenspace corresponding to
λ=1
is
1
1
.
(Use a comma to separate answers as needed.)
5.1.12
Find a basis for the eigenspace corresponding to each listed eigenvalue.
A=
56
22
,
λ=1,
2
A basis for the eigenspace corresponding to
λ=1
is
32
1
.
(Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element. Use a comma to separate answers as needed.)
A basis for the eigenspace corresponding to
λ=2
is
2
1
.
(Type a vector or list of vectors. Type an integer or simplified fraction for each matrix element. Use a comma to separate answers as needed.)
5.1.14
Find a basis for the eigenspace corresponding to the eigenvalue of A given below.
A=
501
301
114
,
λ=4
A basis for the eigenspace corresponding to
λ=4
is
1
1
1
.
(Use a comma to separate answers as needed.)
5.1.14
Find a basis for the eigenspace corresponding to the eigenvalue of A given below.
A=
702
2010
3315
,
λ=6
A basis for the eigenspace corresponding to
λ=6
is
2
1
1
.
(Use a comma to separate answers as needed.)
5.1.17
Find the eigenvalues of the matrix.
000
058
004
The eigenvalue(s) of the matrix is/are
0,4,5.
(Use a comma to separate answers as needed.)
5.1.19
For
A=
124
124
124
,
find one eigenvalue, with no calculation. Justify your answer.
Choose the correct answer below.
One eigenvalue of A is
λ=1.
This is because each row of A is equal to the product of 1 and the row above it.
One eigenvalue of A is
λ=2.
This is because each column of A is equal to the
product
of
2
and the column to the left of it.
One eigenvalue of A is
λ=1.
This is because
1
is one of the entries on the main diagonal of A, which are the eigenvalues of A.
One eigenvalue of A is
λ=0.
This is because the columns of A are linearly dependent, so the matrix is not invertible.
5.1.19
For
A=
3711
3711
3711
,
find one eigenvalue, with no calculation. Justify your answer.
Choose the correct answer below.
One eigenvalue of A is
λ=0.
This is because the columns of A are linearly dependent, so the matrix is not invertible.
One eigenvalue of A is
λ=4.
This is because each column of A is equal to the
sum
of
4
and the column to the left of it.
One eigenvalue of A is
λ=1.
This is because each row of A is equal to the product of 1 and the row above it.
One eigenvalue of A is
λ=3.
This is because
3
is one of the entries on the main diagonal of A, which are the eigenvalues of A.
5.1.21
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If
Ax=λx
for some vector
x,
then
λ
is an eigenvalue of A.
Choose the correct answer below.
The statement is true. If
Ax=λx
for some vector
x,
then
λ
is an eigenvalue of A because the only solution to this equation is the trivial solution.
The statement is false. The equation
Ax=λx
is not used to determine eigenvalues. If
λAx=0
for some vector
x,
then
λ
is an eigenvalue of A.
The statement is false. The condition that
Ax=λx
for some vector
x
is not sufficient to determine if
λ
is an eigenvalue. The equation
Ax=λx
must have a nontrivial solution.
The statement is true. If
Ax=λx
for some vector
x,
then
λ
is an eigenvalue of A by the definition of an eigenvalue.
5.1.21
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If
Ax=λx
for some vector
x,
then
λ
is an eigenvalue of A.
Choose the correct answer below.
The statement is false. The condition that
Ax=λx
for some vector
x
is not sufficient to determine if
λ
is an eigenvalue. The equation
Ax=λx
must have a nontrivial solution.
The statement is true. If
Ax=λx
for some vector
x,
then
λ
is an eigenvalue of A by the definition of an eigenvalue.
The statement is true. If
Ax=λx
for some vector
x,
then
λ
is an eigenvalue of A because the only solution to this equation is the trivial solution.
The statement is false. The equation
Ax=λx
is not used to determine eigenvalues. If
λAx=0
for some vector
x,
then
λ
is an eigenvalue of A.
5.1.22
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If
Ax=λx
for some scalar
λ,
then
x
is an eigenvector of A.
Choose the correct answer below.
The statement is false. The equation
Ax=λx
is not used to determine eigenvectors. If
λAx=0
for some scalar
λ,
then
x
is an eigenvector of A.
The statement is false. The condition that
Ax=λx
for some scalar
λ
is not sufficient to determine if
x
is an eigenvector of A. The vector x must be nonzero.
The statement is true. If
Ax=λx
for some scalar
λ,
then
x
is an eigenvector of A because
λ
is an inverse of A.
The statement is true. If
Ax=λx
for some scalar
λ,
then
x
is an eigenvector of A because the only solution to this equation is the trivial solution.
5.1.23
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
A matrix A is not invertible if and only if 0 is an eigenvalue of A.
Choose the correct answer below.
The statement is false. If 0 is an eigenvalue of A, then there are nontrivial solutions to the equation
Ax=0x.
The equation
Ax=0x
is equivalent to the equation
Ax=0,
and
Ax=0
has nontrivial solutions if and only if A is invertible.
The statement is false. If 0 is an eigenvalue of A, then the equation
Ax=0x
has only the trivial solution. The equation
Ax=0x
is equivalent to the equation
Ax=0,
and
Ax=0
has only the trivial solution if and only if A is invertible.
The statement is true. If 0 is an eigenvalue of A, then the equation
Ax=0x
has only the trivial solution. The equation
Ax=0x
is equivalent to the equation
Ax=0,
and
Ax=0
has only the trivial solution if and only if A is not invertible.
The statement is true. If 0 is an eigenvalue of A, then there are nontrivial solutions to the equation
Ax=0x.
The equation
Ax=0x
is equivalent to the equation
Ax=0,
and
Ax=0
has nontrivial solutions if and only if A is not invertible.
5.1.23
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
A matrix A is not invertible if and only if 0 is an eigenvalue of A.
Choose the correct answer below.
The statement is true. If 0 is an eigenvalue of A, then there are nontrivial solutions to the equation
Ax=0x.
The equation
Ax=0x
is equivalent to the equation
Ax=0,
and
Ax=0
has nontrivial solutions if and only if A is not invertible.
The statement is false. If 0 is an eigenvalue of A, then the equation
Ax=0x
has only the trivial solution. The equation
Ax=0x
is equivalent to the equation
Ax=0,
and
Ax=0
has only the trivial solution if and only if A is invertible.
The statement is false. If 0 is an eigenvalue of A, then there are nontrivial solutions to the equation
Ax=0x.
The equation
Ax=0x
is equivalent to the equation
Ax=0,
and
Ax=0
has nontrivial solutions if and only if A is invertible.
The statement is true. If 0 is an eigenvalue of A, then the equation
Ax=0x
has only the trivial solution. The equation
Ax=0x
is equivalent to the equation
Ax=0,
and
Ax=0
has only the trivial solution if and only if A is not invertible.
5.1.24
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
A number c is an eigenvalue of A if and only if the equation
(AcI)x=0
has a nontrivial solution.
Choose the correct answer below.
The statement is false. A number c is an eigenvalue of A if and only if the equation
Ax=cx
has only the trivial solution, and
Ax=cx
and
(AcI)x=0
are equivalent equations.
The statement is false. A number c is an eigenvalue of A if and only if the equation
Ax=cx
has nontrivial solutions, and the equation
(AcI)x=0
has no bearing on whether c is an eigenvalue.
The statement is true. A number c is an eigenvalue of A if and only if the equation
Ax=cx
has only the trivial solution, and
Ax=cx
and
(AcI)x=0
are inverse equations.
The statement is true. A number c is an eigenvalue of A if and only if the equation
Ax=cx
has nontrivial solutions, and
Ax=cx
and
(AcI)x=0
are equivalent equations.
5.1.24
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
A number c is an eigenvalue of A if and only if the equation
(AcI)x=0
has a nontrivial solution.
Choose the correct answer below.
The statement is true. A number c is an eigenvalue of A if and only if the equation
Ax=cx
has only the trivial solution, and
Ax=cx
and
(AcI)x=0
are inverse equations.
The statement is true. A number c is an eigenvalue of A if and only if the equation
Ax=cx
has nontrivial solutions, and
Ax=cx
and
(AcI)x=0
are equivalent equations.
The statement is false. A number c is an eigenvalue of A if and only if the equation
Ax=cx
has nontrivial solutions, and the equation
(AcI)x=0
has no bearing on whether c is an eigenvalue.
The statement is false. A number c is an eigenvalue of A if and only if the equation
Ax=cx
has only the trivial solution, and
Ax=cx
and
(AcI)x=0
are equivalent equations.
5.1.26
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
To find the eigenvalues of A, reduce A to echelon form.
Choose the correct answer below.
The statement is true. An echelon form of a matrix A displays the eigenvalues as pivots of A.
The statement is false. An echelon form of
A1
displays the eigenvalues of a matrix A.
The statement is true. An echelon form of a matrix A displays the eigenvalues on the main diagonal of A.
The statement is false. An echelon form of a matrix A usually does not display the eigenvalues of A.
5.1.26
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
To find the eigenvalues of A, reduce A to echelon form.
Choose the correct answer below.
The statement is false. An echelon form of
A1
displays the eigenvalues of a matrix A.
The statement is false. An echelon form of a matrix A usually does not display the eigenvalues of A.
The statement is true. An echelon form of a matrix A displays the eigenvalues as pivots of A.
The statement is true. An echelon form of a matrix A displays the eigenvalues on the main diagonal of A.
5.1.27
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If
v1
and
v2
are linearly independent eigenvectors, then they correspond to distinct eigenvalues.
Choose the correct answer below.
The statement is true. If
v1
and
v2
are linearly independent eigenvectors, then they correspond to distinct eigenvalues, because eigenvectors must be nonzero.
The statement is true. If
v1
and
v2
are linearly independent eigenvectors, then they correspond to distinct eigenvalues, because all eigenvectors of a single eigenvalue are linearly dependent.
The statement is false.
There
may be linearly independent eigenvectors that both correspond to the same eigenvalue.
The statement is false. Every eigenvalue has an infinite number of corresponding eigenvectors.
5.1.28
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The eigenvalues of a matrix are on its main diagonal.
Choose the correct answer below.
The statement is true. The eigenvalues of a matrix are on its main diagonal because the main diagonal remains the same when the matrix is transposed, and a matrix and its transpose have the same eigenvalues.
The statement is false. The matrix must first be reduced to echelon form. The eigenvalues are on the main diagonal of the echelon form of the matrix.
The statement is true. The eigenvalues of a matrix are on its main diagonal because the main diagonal determines the pivots of the matrix, which are used to calculate the eigenvalues.
The statement is false. If the matrix is a triangular matrix, the values on the main diagonal are eigenvalues. Otherwise, the main diagonal may or may not contain eigenvalues.
5.1.30
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
An eigenspace of A is a null space of a certain matrix.
Choose the correct answer below.
The statement is true. An eigenspace of A corresponding to the eigenvalue
λ
is the null space of the matrix
(AλI).
The statement is false. An eigenspace of A is not a null space of a certain matrix because an eigenspace consists of all solutions x to the equation
Ax=λb,
which does not include the zero vector unless
b=0.
The statement is true. An eigenspace of A corresponding to the eigenvalue
λ
is the null space of the matrix
(λAI).
The statement is false. An eigenspace of A is not a null space of a certain matrix because an eigenspace consists of all the eigenvectors corresponding to an eigenvalue
λ,
and eigenvectors are by definition nonzero vectors, so the eigenspace does not include the zero
vector.
5.1.30
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
An eigenspace of A is a null space of a certain matrix.
Choose the correct answer below.
The statement is true. An eigenspace of A corresponding to the eigenvalue
λ
is the null space of the matrix
(λAI).
The statement is false. An eigenspace of A is not a null space of a certain matrix because an eigenspace consists of all the eigenvectors corresponding to an eigenvalue
λ,
and eigenvectors are by definition nonzero vectors, so the eigenspace does not include the zero
vector.
The statement is false. An eigenspace of A is not a null space of a certain matrix because an eigenspace consists of all solutions x to the equation
Ax=λb,
which does not include the zero vector unless
b=0.
The statement is true. An eigenspace of A corresponding to the eigenvalue
λ
is the null space of the matrix
(AλI).
5.1.30
A is an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
An eigenspace of A is a null space of a certain matrix.
Choose the correct answer below.
The statement is true. An eigenspace of A corresponding to the eigenvalue
λ
is the null space of the matrix
(AλI).
The statement is false. An eigenspace of A is not a null space of a certain matrix because an eigenspace consists of all the eigenvectors corresponding to an eigenvalue
λ,
and eigenvectors are by definition nonzero vectors, so the eigenspace does not include the zero
vector.
The statement is false. An eigenspace of A is not a null space of a certain matrix because an eigenspace consists of all solutions x to the equation
Ax=λb,
which does not include the zero vector unless
b=0.
The statement is true. An eigenspace of A corresponding to the eigenvalue
λ
is the null space of the matrix
(λAI).
5.2.1
Find the characteristic polynomial and the eigenvalues of the matrix.
75
57
The characteristic polynomial is
λ214λ+24.
(Type an expression using
λ
as the variable. Type an exact answer, using radicals as needed.)
Select the correct choice below and, if necessary, fill in the answer box within your choice.
The real eigenvalue(s) of the matrix is/are
2,12.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
The matrix has no real eigenvalues.
5.2.1
Find the characteristic polynomial and the eigenvalues of the matrix.
102
210
The characteristic polynomial is
λ220λ+96.
(Type an expression using
λ
as the variable. Type an exact answer, using radicals as needed.)
Select the correct choice below and, if necessary, fill in the answer box within your choice.
The real eigenvalue(s) of the matrix is/are
8,12.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
The matrix has no real eigenvalues.
5.2.1
Find the characteristic polynomial and the eigenvalues of the matrix.
75
57
The characteristic polynomial is
λ214λ+24.
(Type an expression using
λ
as the variable. Type an exact answer, using radicals as needed.)
Select the correct choice below and, if necessary, fill in the answer box within your choice.
The real eigenvalue(s) of the matrix is/are
2,12.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
The matrix has no real eigenvalues.
5.2.6
Find the characteristic polynomial and the eigenvalues of the matrix.
76
62
The characteristic polynomial is
λ29λ+50.
(Type an expression using
λ
as the variable. Type an exact answer, using radicals as needed.)
Select the correct choice below and, if necessary, fill in the answer box within your choice.
The real eigenvalue(s) of the matrix is/are
.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
The matrix has no real eigenvalues.
5.2.7
Find the characteristic polynomial and the eigenvalues of the matrix.
52
77
The characteristic polynomial is
λ2+12λ+49.
(Type an expression using
λ
as the variable. Type an exact answer, using radicals as needed.)
Select the correct choice below and, if necessary, fill in the answer box within your choice.
The real eigenvalue(s) of the matrix is/are
.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
The matrix has no real eigenvalues.
5.2.7
Find the characteristic polynomial and the eigenvalues of the matrix.
92
55
The characteristic polynomial is
λ214λ+55.
(Type an expression using
λ
as the variable. Type an exact answer, using radicals as needed.)
Select the correct choice below and, if necessary, fill in the answer box within your choice.
The real eigenvalue(s) of the matrix is/are
.
(Type an exact answer, using radicals as needed. Use a comma to separate answers as needed. Type each answer only once.)
The matrix has no real eigenvalues.
5.2.9
Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for
3×3
determinants. [Note: Finding the characteristic polynomial of a
3×3
matrix is not easy to do with just row operations, because the variable
λ
is involved.]
101
524
050
The characteristic polynomial is
λ3+3λ222λ5.
(Type an expression using
λ
as the variable.)
5.2.9
Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for
3×3
determinants. [Note: Finding the characteristic polynomial of a
3×3
matrix is not easy to do with just row operations, because the variable
λ
is involved.]
101
524
070
The characteristic polynomial is
λ3+3λ230λ7.
(Type an expression using
λ
as the variable.)
5.2.9
Find the characteristic polynomial of the matrix, using either a cofactor expansion or the special formula for
3×3
determinants. [Note: Finding the characteristic polynomial of a
3×3
matrix is not easy to do with just row operations, because the variable
λ
is involved.]
101
524
070
The characteristic polynomial is
λ3+3λ230λ7.
(Type an expression using
λ
as the variable.)
5.2.18
It can be shown that the algebraic multiplicity of an eigenvalue
λ
is always greater than or equal to the dimension of the eigenspace corresponding to
λ.
Find h in the matrix A below such that the eigenspace for
λ=5
is two-dimensional.
A=
5283
03h0
0057
0001
The value of h for which the eigenspace for
λ=5
is two-dimensional is
h=8.
5.2.18
It can be shown that the algebraic multiplicity of an eigenvalue
λ
is always greater than or equal to the dimension of the eigenspace corresponding to
λ.
Find h in the matrix A below such that the eigenspace for
λ=7
is two-dimensional.
A=
7485
03h0
0074
0004
The value of h for which the eigenspace for
λ=7
is two-dimensional is
h=8.
5.2.18
It can be shown that the algebraic multiplicity of an eigenvalue
λ
is always greater than or equal to the dimension of the eigenspace corresponding to
λ.
Find h in the matrix A below such that the eigenspace for
λ=8
is two-dimensional.
A=
8365
05h0
0083
0001
The value of h for which the eigenspace for
λ=8
is two-dimensional is
h=6.
5.2.27
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If
λ+5
is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.
Choose the correct answer below.
The statement is true. If
λ+5
is a factor of the characteristic polynomial of A, then
λ+5
must be an entry on the main diagonal of
AλI.
The statement is false. If
λ+5
is a factor of the characteristic polynomial of A, then
5
is an eigenvalue of A. In order for 5 to be an eigenvalue of A, the characteristic polynomial would need to have a factor of
λ5.
The statement is false. The factors of the characteristic polynomial are useful for determining the eigenvectors of a matrix, not its eigenvalues.
The statement is true. If
λ+5
is a factor of the characteristic polynomial of A, then
λ=5
is a root of the characteristic equation.
5.2.27
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If
λ+5
is a factor of the characteristic polynomial of A, then 5 is an eigenvalue of A.
Choose the correct answer below.
The statement is true. If
λ+5
is a factor of the characteristic polynomial of A, then
λ+5
must be an entry on the main diagonal of
AλI.
The statement is true. If
λ+5
is a factor of the characteristic polynomial of A, then
λ=5
is a root of the characteristic equation.
The statement is false. The factors of the characteristic polynomial are useful for determining the eigenvectors of a matrix, not its eigenvalues.
The statement is false. If
λ+5
is a factor of the characteristic polynomial of A, then
5
is an eigenvalue of A. In order for 5 to be an eigenvalue of A, the characteristic polynomial would need to have a factor of
λ5.
5.2.28
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
The multiplicity of a root r of the characteristic equation of A is called the algebraic multiplicity of r as an eigenvalue of A.
Choose the correct answer below.
The statement is true. It is the definition of the algebraic multiplicity of an eigenvalue of A.
The statement is false. The multiplicity of a root r of the characteristic equation of A is the number of eigenvectors corresponding to that root.
The statement is true. It is the definition of the multiplicity of a root of the characteristic equation of A.
The statement is false. The multiplicity of a root r of the characteristic equation of A is called the geometric multiplicity of r as an eigenvalue of A.
5.3.1
Let
A=PDP1
and P and D as shown below. Compute
A4.
P=
12
23
,
D=
10
02
A4=
6130
9044
(Simplify your answer.)
5.3.1
Let
A=PDP1
and P and D as shown below. Compute
A4.
P=
14
27
,
D=
10
03
A4=
641320
1120559
(Simplify your answer.)
5.3.1
Let
A=PDP1
and P and D as shown below. Compute
A4.
P=
14
27
,
D=
10
03
A4=
641320
1120559
(Simplify your answer.)
5.3.5
Matrix A is factored in the form
PDP1.
Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.
A=
311
131
113
=
112
102
110
500
020
002
131313
131323
161316
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
There is one distinct eigenvalue,
λ=.
A basis for the corresponding eigenspace is
.
In ascending order, the two distinct eigenvalues are
λ1=2
and
λ2=5.
Bases for the corresponding eigenspaces are
1
0
1
,
2
2
0
and
1
1
1
,
respectively.
In ascending order, the three distinct eigenvalues are
λ1=,
λ2=,
and
λ3=.
Bases for the corresponding eigenspaces are
,
,
and
,
respectively.
5.3.5
Matrix A is factored in the form
PDP1.
Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.
A=
211
232
112
=
122
202
120
500
010
001
141414
181838
141414
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
There is one distinct eigenvalue,
λ=.
A basis for the corresponding eigenspace is
.
In ascending order, the two distinct eigenvalues are
λ1=1
and
λ2=5.
Bases for the corresponding eigenspaces are
2
0
2
,
2
2
0
and
1
2
1
,
respectively.
In ascending order, the three distinct eigenvalues are
λ1=,
λ2=,
and
λ3=.
Bases for the corresponding eigenspaces are
,
,
and
,
respectively.
5.3.5
Matrix A is factored in the form
PDP1.
Use the Diagonalization Theorem to find the eigenvalues of A and a basis for each eigenspace.
A=
221
131
122
=
212
201
210
500
010
001
181418
141234
141214
Select the correct choice below and fill in the answer boxes to complete your choice.
(Use a comma to separate vectors as needed.)
There is one distinct eigenvalue,
λ=.
A basis for the corresponding eigenspace is
.
In ascending order, the two distinct eigenvalues are
λ1=1
and
λ2=5.
Bases for the corresponding eigenspaces are
1
0
1
,
2
1
0
and
2
2
2
,
respectively.
In ascending order, the three distinct eigenvalues are
λ1=,
λ2=,
and
λ3=.
Bases for the corresponding eigenspaces are
,
,
and
,
respectively.
5.3.7
Diagonalize the following matrix, if possible.
10
81
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=,
D=
10
04
For
P=
10
41
,
D=
10
01
For
P=,
D=
80
08
The matrix cannot be diagonalized.
5.3.7
Diagonalize the following matrix, if possible.
70
87
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=
70
41
,
D=
70
07
For
P=,
D=
70
04
For
P=,
D=
80
08
The matrix cannot be diagonalized.
5.3.7
Diagonalize the following matrix, if possible.
80
108
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=
80
51
,
D=
80
08
For
P=,
D=
100
010
For
P=,
D=
80
05
The matrix cannot be diagonalized.
5.3.8
Diagonalize the following matrix.
51
05
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=,
D=
10
01
.
(Type an integer or simplified fraction for each matrix element.)
For
P=,
D=
50
05
.
(Type an integer or simplified fraction for each matrix element.)
For
P=,
D=
50
01
.
(Type an integer or simplified fraction for each matrix element.)
The matrix cannot be diagonalized.
5.3.8
Diagonalize the following matrix.
181
018
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=,
D=
10
01
.
(Type an integer or simplified fraction for each matrix element.)
For
P=,
D=
180
01
.
(Type an integer or simplified fraction for each matrix element.)
For
P=,
D=
180
018
.
(Type an integer or simplified fraction for each matrix element.)
The matrix cannot be diagonalized.
5.3.11
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.
101726
62328
61823
;
λ=1,4,5
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=
132
124
123
,
D=
100
040
005
.
(Simplify your answer.)
The matrix cannot be diagonalized.
5.3.11
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.
5148
9166
9111
;
λ=1,4,5
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=
121
133
134
,
D=
100
040
005
.
(Simplify your answer.)
The matrix cannot be diagonalized.
5.3.11
Diagonalize the following matrix. The real eigenvalues are given to the right of the matrix.
81218
41820
41315
;
λ=2,4,5
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=
132
124
123
,
D=
200
040
005
.
(Simplify your answer.)
The matrix cannot be diagonalized.
5.3.18
Diagonalize the following matrix. One eigenvalue is
λ=2
and one eigenvector is (3,3,3).
461
241
263
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=,
D=
100
010
002
.
(Type an integer or simplified fraction for each matrix element.)
For
P=
161
120
102
,
D=
100
020
002
.
(Type an integer or simplified fraction for each matrix element.)
The matrix cannot be diagonalized.
5.3.18
Diagonalize the following matrix. One eigenvalue is
λ=6
and one eigenvector is (1,1,2).
174411
113811
228828
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=
141
110
201
,
D=
500
060
006
.
(Type an integer or simplified fraction for each matrix element.)
For
P=,
D=
2200
0220
006
.
(Type an integer or simplified fraction for each matrix element.)
The matrix cannot be diagonalized.
5.3.18
Diagonalize the following matrix. One eigenvalue is
λ=4
and one eigenvector is (3,2,2).
8123
8122
882
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
For
P=
311
201
240
,
D=
200
040
004
.
(Type an integer or simplified fraction for each matrix element.)
For
P=,
D=
200
020
004
.
(Type an integer or simplified fraction for each matrix element.)
The matrix cannot be diagonalized.
5.3.21
Let A, P, and D be
n×n
matrices. Determine whether the statement below is true or false. Justify the answer.
A is diagonalizable if
A=PDP1
for some matrix D and some invertible matrix P.
Choose the correct answer below.
The statement is false. A must have n distinct eigenvalues for the matrix D and P to exist.
The statement is false. The symbol D does not automatically denote a diagonal matrix.
The statement is true. If
A=PDP1,
then A is diagonalizable by definition.
The statement is true. The columns of P are the n linearly independent eigenvectors of A.
5.3.21
Let A, P, and D be
n×n
matrices. Determine whether the statement below is true or false. Justify the answer.
A is diagonalizable if
A=PDP1
for some matrix D and some invertible matrix P.
Choose the correct answer below.
The statement is false. A must have n distinct eigenvalues for the matrix D and P to exist.
The statement is true. If
A=PDP1,
then A is diagonalizable by definition.
The statement is false. The symbol D does not automatically denote a diagonal matrix.
The statement is true. The columns of P are the n linearly independent eigenvectors of A.
5.3.22
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If
n
has a basis of eigenvectors of A, then A is diagonalizable.
Choose the correct answer below.
The statement is true. All square matrices are diagonalizable.
The statement is false. The eigenvectors of A must also be linearly independent.
The statement is false. A must also have n distinct eigenvalues.
The statement is true. A is diagonalizable if and only if there are enough eigenvectors to form a basis of
n.
5.3.22
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If
n
has a basis of eigenvectors of A, then A is diagonalizable.
Choose the correct answer below.
The statement is false. A must also have n distinct eigenvalues.
The statement is true. All square matrices are diagonalizable.
The statement is true. A is diagonalizable if and only if there are enough eigenvectors to form a basis of
n.
The statement is false. The eigenvectors of A must also be linearly independent.
5.3.23
Let A, P, and D be
n×n
matrices. Determine whether the statement below is true or false. Justify the answer.
A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.
Choose the correct answer below.
The statement is true.
A=PDP1,
where the diagonal entries of D are eigenvalues of A that correspond, respectively, to the eigenvectors in P.
The statement is true. There are n corresponding eigenvectors that form a basis of
n.
The statement is false. The eigenvalues of A may not produce enough eigenvectors to form a basis of
n.
The statement is false. It also required that A is invertible.
5.3.23
Let A, P, and D be
n×n
matrices. Determine whether the statement below is true or false. Justify the answer.
A is diagonalizable if and only if A has n eigenvalues, counting multiplicities.
Choose the correct answer below.
The statement is true.
A=PDP1,
where the diagonal entries of D are eigenvalues of A that correspond, respectively, to the eigenvectors in P.
The statement is false. It also required that A is invertible.
The statement is false. The eigenvalues of A may not produce enough eigenvectors to form a basis of
n.
The statement is true. There are n corresponding eigenvectors that form a basis of
n.
5.3.24
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If A is diagonalizable, then A is invertible.
Choose the correct answer below.
The statement is false. Invertibility depends on 0 not being an eigenvalue. A diagonalizable matrix may or may not have 0 as an eigenvalue.
The statement is true. If A is invertible, then it has n distinct eigenvectors that form a basis of
n.
The statement is true. If A is diagonalizable, then det(A) does not equal 0. Thus, A is invertible.
The statement is false. If A is diagonalizable, then
det(AλI)=0
has a
solution.
Thus, A is not invertible.
5.3.24
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If A is diagonalizable, then A is invertible.
Choose the correct answer below.
The statement is false. If A is diagonalizable, then
det(AλI)=0
has a
solution.
Thus, A is not invertible.
The statement is true. If A is diagonalizable, then det(A) does not equal 0. Thus, A is invertible.
The statement is true. If A is invertible, then it has n distinct eigenvectors that form a basis of
n.
The statement is false. Invertibility depends on 0 not being an eigenvalue. A diagonalizable matrix may or may not have 0 as an eigenvalue.
5.3.25
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
A matrix A is diagonalizable if A has n eigenvectors.
Choose the correct answer below.
The statement is false. A diagonalizable matrix must have n linearly independent eigenvectors.
The statement is true. A diagonalizable matrix must have a minimum of n linearly independent eigenvectors.
The statement is true. A diagonalizable matrix must have more than one linearly independent eigenvector.
The statement is false. A matrix is diagonalizable if and only if it has
n1
linearly independent eigenvectors.
5.3.26
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If A is diagonalizable, then A has n distinct eigenvalues.
Choose the correct answer below.
The statement is true. A diagonalizable matrix must have exactly n eigenvalues.
The statement is false. A diagonalizable matrix must have more than n eigenvalues.
The statement is false. A diagonalizable matrix can have fewer than n eigenvalues and still have n linearly independent eigenvectors.
The statement is true. A diagonalizable matrix must have n distinct eigenvalues.
5.3.28
Let A be an
n×n
matrix. Determine whether the statement below is true or false. Justify the answer.
If A is invertible, then A is diagonalizable.
Choose the correct answer below.
The statement is false. An invertible matrix may have fewer than n linearly independent eigenvectors, making it not diagonalizable.
The statement is true. If a matrix is invertible, then it has n linearly independent eigenvectors, making it diagonalizable.
The statement is false.
Invertible
matrices always have a maximum of n linearly independent eigenvectors, making it not diagonalizable.
The statement is true. A diagonalizable matrix is invertible, so an invertible matrix is diagonalizable.
5.3.30
A is a
3×3
matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable? Why?
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
No. A matrix with 3 columns must have
unique eigenvalues in order to be diagonalizable.
No. The sum of the dimensions of the eigenspaces equals
2
and the matrix has 3 columns. The sum of the dimensions of the eigenspace and the number of columns must be equal.
Yes. One of the eigenspaces would have
unique eigenvectors. Since the eigenvector for the third eigenvalue would also be unique, A must be diagonalizable.
Yes. As long as the collection of eigenvectors spans
3,
A is diagonalizable.
5.3.30
A is a
3×3
matrix with two eigenvalues. Each eigenspace is one-dimensional. Is A diagonalizable? Why?
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
Yes. One of the eigenspaces would have
unique eigenvectors. Since the eigenvector for the third eigenvalue would also be unique, A must be diagonalizable.
No. A matrix with 3 columns must have
unique eigenvalues in order to be diagonalizable.
No. The sum of the dimensions of the eigenspaces equals
2
and the matrix has 3 columns. The sum of the dimensions of the eigenspace and the number of columns must be equal.
Yes. As long as the collection of eigenvectors spans
3,
A is diagonalizable.
5.3.35
A factorization
A=PDP1
is not unique. For
A=
92
121
,
one factorization is
P=
11
23
,
D=
50
03
,
and
P1=
31
21
.
Use this information with
D1=
30
05
to find a matrix
P1
such that
A=P1D1P11.
P1=
11
32
(Type an integer or simplified fraction for each matrix element.)
5.3.35
A factorization
A=PDP1
is not unique. For
A=
61
61
,
one factorization is
P=
11
23
,
D=
40
03
,
and
P1=
31
21
.
Use this information with
D1=
30
04
to find a matrix
P1
such that
A=P1D1P11.
P1=
11
32
(Type an integer or simplified fraction for each matrix element.)
5.3.35
A factorization
A=PDP1
is not unique. For
A=
92
121
,
one factorization is
P=
11
23
,
D=
50
03
,
and
P1=
31
21
.
Use this information with
D1=
30
05
to find a matrix
P1
such that
A=P1D1P11.
P1=
11
32
(Type an integer or simplified fraction for each matrix element.)
5.3.37
Identify a nonzero
2×2
matrix that is invertible but not diagonalizable.
Choose the correct answer below.
11
01
10
00
11
11
10
10
11
00
10
01
5.3.37
Identify a nonzero
2×2
matrix that is invertible but not diagonalizable.
Choose the correct answer below.
11
00
11
11
10
01
11
01
10
00
10
10
5.3.37
Identify a nonzero
2×2
matrix that is invertible but not diagonalizable.
Choose the correct answer below.
10
01
10
10
11
11
11
00
11
01
10
00
5.5.1
Let the matrix below act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
117
19
The eigenvalues of
117
19
are
5±i.
(Type an exact answer, using radicals and
i
as needed. Use a comma to separate answers as needed.)
Find a basis for the eigenspace corresponding to the eigenvalue
a+bi,
where
b>0.
Choose the correct answer below.
4+i
1
i
1
1
4+i
4i
1
1
4i
i
1
Find a basis for the eigenspace corresponding to the eigenvalue
abi,
where
b>0.
Choose the correct answer below.
4+i
1
i
1
4i
1
i
1
1
4i
1
4+i
5.5.1
Let the matrix below act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
117
19
The eigenvalues of
117
19
are
5±i.
(Type an exact answer, using radicals and
i
as needed. Use a comma to separate answers as needed.)
Find a basis for the eigenspace corresponding to the eigenvalue
a+bi,
where
b>0.
Choose the correct answer below.
i
1
1
4i
4+i
1
1
4+i
4i
1
i
1
Find a basis for the eigenspace corresponding to the eigenvalue
abi,
where
b>0.
Choose the correct answer below.
i
1
1
4i
i
1
1
4+i
4+i
1
4i
1
5.5.1
Let the matrix below act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
126
111
The eigenvalues of
126
111
are
6±i.
(Type an exact answer, using radicals and
i
as needed. Use a comma to separate answers as needed.)
Find a basis for the eigenspace corresponding to the eigenvalue
a+bi,
where
b>0.
Choose the correct answer below.
i
1
5i
1
5+i
1
1
5+i
1
5i
i
1
Find a basis for the eigenspace corresponding to the eigenvalue
abi,
where
b>0.
Choose the correct answer below.
1
5+i
5i
1
1
5i
i
1
5+i
1
i
1
5.5.3
Let the matrix
21
458
act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
Select all that apply.
λ=5+6i;
v=
36i
45
λ=5+6i;
v=
36i
6
λ=56i;
v=
36i
6
λ=56i;
v=
3+6i
45
λ=6+5i;
v=
36i
45
λ=5+6i;
v=
3+6i
5
λ=65i;
v=
3+6i
45
λ=56i;
v=
3+6i
5
5.5.3
Let the matrix
21
258
act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
Select all that apply.
λ=54i;
v=
34i
4
λ=5+4i;
v=
3+4i
5
λ=4+5i;
v=
34i
25
λ=5+4i;
v=
34i
25
λ=54i;
v=
3+4i
25
λ=45i;
v=
3+4i
25
λ=5+4i;
v=
34i
4
λ=54i;
v=
3+4i
5
5.5.3
Let the matrix
61
174
act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
Select all that apply.
λ=5+4i;
v=
1+4i
5
λ=54i;
v=
1+4i
5
λ=54i;
v=
14i
4
λ=4+5i;
v=
14i
17
λ=45i;
v=
1+4i
17
λ=5+4i;
v=
14i
17
λ=54i;
v=
1+4i
17
λ=5+4i;
v=
14i
4
5.5.5
Let the matrix below act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
01
16218
The eigenvalues of
01
16218
are
9±9i.
(Type an exact answer, using radicals and
i
as needed. Use a comma to separate answers as needed.)
Find a basis for the eigenspace corresponding to the eigenvalue
a+bi.
Choose the correct answer below.
1
9+9i
1
9+i
9+i
1
9i
1
1
9i
9+9i
1
Find a basis for the eigenspace corresponding to the eigenvalue
abi.
Choose the correct answer below.
1
9i
9i
1
9i
1
99i
1
1
99i
1
9i
5.5.6
Let the matrix below act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
22
22
The eigenvalues of
22
22
are
2±2i.
(Type an exact answer, using radicals and
i
as needed. Use a comma to separate answers as needed.)
A basis for the eigenspace corresponding to the eigenvalue
a+bi,
where
b>0,
is
i
1
.
(Type an exact answer, using radicals and
i
as needed.)
A basis for the eigenspace corresponding to the eigenvalue
abi
where
b>0,
is
i
1
.
(Type an exact answer, using radicals and
i
as needed.)
5.5.6
Let the matrix below act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
54
45
The eigenvalues of
54
45
are
5±4i.
(Type an exact answer, using radicals and
i
as needed. Use a comma to separate answers as needed.)
A basis for the eigenspace corresponding to the eigenvalue
a+bi,
where
b>0,
is
i
1
.
(Type an exact answer, using radicals and
i
as needed.)
A basis for the eigenspace corresponding to the eigenvalue
abi
where
b>0,
is
i
1
.
(Type an exact answer, using radicals and
i
as needed.)
5.5.6
Let the matrix below act on
2.
Find the eigenvalues and a basis for each eigenspace in
2.
13
31
The eigenvalues of
13
31
are
1±3i.
(Type an exact answer, using radicals and
i
as needed. Use a comma to separate answers as needed.)
A basis for the eigenspace corresponding to the eigenvalue
a+bi,
where
b>0,
is
i
1
.
(Type an exact answer, using radicals and
i
as needed.)
A basis for the eigenspace corresponding to the eigenvalue
abi
where
b>0,
is
i
1
.
(Type an exact answer, using radicals and
i
as needed.)
5.5.7
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
43  4 
 4 43 
The eigenvalues of A are
λ=43±4i.
(Simplify your answer. Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=π6
(Simplify your answer. Type an exact answer, using
π
as needed.)
r=8
(Simplify your answer.)
5.5.7
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
3  1 
 1 3 
The eigenvalues of A are
λ=3±i.
(Simplify your answer. Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=5π6
(Simplify your answer. Type an exact answer, using
π
as needed.)
r=2
(Simplify your answer.)
5.5.7
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
33  3 
 3 33 
The eigenvalues of A are
λ=33±3i.
(Simplify your answer. Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=π6
(Simplify your answer. Type an exact answer, using
π
as needed.)
r=6
(Simplify your answer.)
5.5.9
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
3313
1333
The eigenvalues of A are
λ=33±13i.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=5π6
(Type an exact answer, using
π
as needed.)
r=23
(Type an exact answer, using radicals as needed.)
5.5.9
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
3414
1434
The eigenvalues of A are
λ=34±14i.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=5π6
(Type an exact answer, using
π
as needed.)
r=12
(Type an exact answer, using radicals as needed.)
5.5.9
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
3717
1737
The eigenvalues of A are
λ=37±17i.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=5π6
(Type an exact answer, using
π
as needed.)
r=27
(Type an exact answer, using radicals as needed.)
5.5.10
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
22
22
The eigenvalues of A are
λ=2±2i.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=π4
(Type an exact answer, using
π
as needed.)
r=22
(Type an exact answer, using radicals as needed.)
5.5.10
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
44
44
The eigenvalues of A are
λ=4±4i.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=3π4
(Type an exact answer, using
π
as needed.)
r=42
(Type an exact answer, using radicals as needed.)
5.5.10
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
66
66
The eigenvalues of A are
λ=6±6i.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=3π4
(Type an exact answer, using
π
as needed.)
r=122
(Type an exact answer, using radicals as needed.)
5.5.11
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
0.80.8
0.80.8
The eigenvalues of A are
λ=45±45i.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=3π4
(Type an exact answer, using
π
as needed.)
r=425
(Type an exact answer, using radicals as needed.)
5.5.12
List the eigenvalues of
A.
The transformation
xAx
is the composition of a rotation and a scaling. Give the angle
φ
of the rotation, where
π<φπ,
and give the scale factor r.
A=
00.8
0.80
The eigenvalues of A are
λ=±45i.
(Use a comma to separate answers as needed. Type an exact answer, using radicals and
i
as needed.)
φ=π2
(Simplify your answer. Type an exact answer, using
π
as needed.)
r=45
(Simplify your answer. Type an exact answer, using radicals as needed.)
5.5.13
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
15
15
has the form
A=PCP1.
The eigenvalues of A are
3i
and
3+i.
The corresponding eigenvectors are
v1=
2i
1
and
v2=
2+i
1
,
respectively.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
21
10
,
31
13
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.13
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
110
17
has the form
A=PCP1.
The eigenvalues of A are
4i
and
4+i.
The corresponding eigenvectors are
v1=
3i
1
and
v2=
3+i
1
,
respectively.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
31
10
,
41
14
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.13
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
12
13
has the form
A=PCP1.
The eigenvalues of A are
2i
and
2+i.
The corresponding eigenvectors are
v1=
1i
1
and
v2=
1+i
1
,
respectively.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
11
10
,
21
12
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.16
Find an invertible matrix P and a matrix C of the form
ab
ba
such that the matrix
A=
237
14
has the form
A=PCP1.
The eigenvalues of A are
3+6i
and
36i
with corresponding eigenvectors
6+i
i
and
6i
i
.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
61
01
,
36
63
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.16
Find an invertible matrix P and a matrix C of the form
ab
ba
such that the matrix
A=
55
17
has the form
A=PCP1.
The eigenvalues of A are
6+2i
and
62i
with corresponding eigenvectors
2+i
i
and
2i
i
.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
21
01
,
62
26
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.16
Find an invertible matrix P and a matrix C of the form
ab
ba
such that the matrix
A=
45
12
has the form
A=PCP1.
The eigenvalues of A are
3+2i
and
32i
with corresponding eigenvectors
2i
i
and
2+i
i
.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
21
01
,
32
23
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.17
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
10.75
2.41.4
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
1214
10
,
1535
3515
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.17
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
12
0.81.4
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
3212
10
,
1525
2515
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.17
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
0.80.6
31.6
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
2515
10
,
2535
3525
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.18
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
0.60.08
10.2
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
1515
10
,
2515
1525
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.18
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
1.20.8
1.60.4
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
1212
10
,
2545
4525
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.18
Find an invertible matrix P and a matrix C of the form
ab
ba
such that
A=
1.20.8
1.60.4
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
1212
10
,
2545
4525
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.19
Find an invertible matrix P and a matrix C of the form
ab
ba
such that the matrix
A=
1.520.7
0.560.4
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
112
10
,
2425725
7252425
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.19
Find an invertible matrix P and a matrix C of the form
ab
ba
such that the matrix
A=
1.20.5
1.040.4
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
5131526
10
,
4535
3545
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
5.5.19
Find an invertible matrix P and a matrix C of the form
ab
ba
such that the matrix
A=
10.5
0.80.6
has the form
A=PCP1.
Select the correct choice below and, if necessary, fill in the answer box to complete your choice.
The matrices P and C are
1434
10
,
4535
3545
.
(Use a comma to separate answers as needed.)
There is no matrix C of the form
ab
ba
and no invertible matrix P such that
A=PCP1.
6.1.1
Compute
uu,
vu,
and
vuuu
using the vectors
u=
2
5
and
v=
7
6
.
uu=29
(Simplify your answer.)
vu=16
(Simplify your answer.)
vuuu=1629
(Type an integer or a simplified fraction.)
6.1.1
Compute
uu,
vu,
and
vuuu
using the vectors
u=
1
4
and
v=
5
3
.
uu=17
(Simplify your answer.)
vu=7
(Simplify your answer.)
vuuu=717
(Type an integer or a simplified fraction.)
6.1.1
Compute
uu,
vu,
and
vuuu
using the vectors
u=
2
3
and
v=
6
7
.
uu=13
(Simplify your answer.)
vu=9
(Simplify your answer.)
vuuu=913
(Type an integer or a simplified fraction.)
6.1.2
Compute
ww,
xw,
and
xwww
using the vectors
w=
3
4
5
and
x=
6
3
3
.
ww=50
(Simplify your answer. Type an integer or a simplified fraction.)
xw=15
(Simplify your answer. Type an integer or a simplified fraction.)
xwww=310
(Simplify your answer. Type an integer or a simplified fraction.)
6.1.2
Compute
ww,
xw,
and
xwww
using the vectors
w=
3
4
5
and
x=
7
3
2
.
ww=50
(Simplify your answer. Type an integer or a simplified fraction.)
xw=23
(Simplify your answer. Type an integer or a simplified fraction.)
xwww=2350
(Simplify your answer. Type an integer or a simplified fraction.)
6.1.4
Compute
1uuu,
where
u=
4
5
.
1uuu=
441
541
(Type an integer or simplified fraction for each matrix element.)
6.1.4
Compute
1uuu,
where
u=
2
3
.
1uuu=
213
313
(Type an integer or simplified fraction for each matrix element.)
6.1.7
Compute
w
using
w=
5
1
5
.
w=51
(Type an exact answer, using radicals as needed.)
6.1.7
Compute
w
using
w=
2
2
7
.
w=57
(Type an exact answer, using radicals as needed.)
6.1.9
Find a unit vector in the direction of the given vector.
72
320
A unit vector in the direction of the given vector is
941
4041
.
(Type an exact answer, using radicals as needed.)
6.1.9
Find a unit vector in the direction of the given vector.
77
264
A unit vector in the direction of the given vector is
77275
264275
.
(Type an exact answer, using radicals as needed.)
6.1.10
Find a unit vector in the direction of the given vector.
2
18
2
A unit vector in the direction of the given vector is
183
983
183
.
(Type an exact answer, using radicals as needed.)
6.1.10
Find a unit vector in the direction of the given vector.
3
18
3
A unit vector in the direction of the given vector is
138
638
138
.
(Type an exact answer, using radicals as needed.)
6.1.13
Find the distance between
x=
12
3
and
y=
4
9
.
The distance between x and y is
273.
(Type an exact answer, using radicals as needed.)
6.1.13
Find the distance between
x=
10
4
and
y=
1
9
.
The distance between x and y is
146.
(Type an exact answer, using radicals as needed.)
6.1.13
Find the distance between
x=
11
3
and
y=
1
9
.
The distance between x and y is
180.
(Type an exact answer, using radicals as needed.)
6.1.14
Find the distance between
u=
0
6
3
and
z=
2
1
6
.
The distance between u and z is
38.
(Type an exact answer, using radicals as needed.)
6.1.14
Find the distance between
u=
0
6
2
and
z=
3
1
5
.
The distance between u and z is
43.
(Type an exact answer, using radicals as needed.)
6.1.15
Determine if the following vectors are orthogonal.
a=
8
3
,
b=
3
2
Select the correct choice below and fill in the answer box to complete your choice.
(Type an integer or a simplified fraction.)
The vectors a and b are not orthogonal because
a+b=.
The vectors a and b are not orthogonal because
ab=18.
The vectors a and b are orthogonal because
ab=.
The vectors a and b are orthogonal because
a+b=.
6.1.15
Determine if the following vectors are orthogonal.
a=
8
3
,
b=
2
2
Select the correct choice below and fill in the answer box to complete your choice.
(Type an integer or a simplified fraction.)
The vectors a and b are orthogonal because
a+b=.
The vectors a and b are not orthogonal because
ab=10.
The vectors a and b are orthogonal because
ab=.
The vectors a and b are not orthogonal because
a+b=.
6.1.16
Determine if the following vectors are orthogonal.
u=
10
4
2
,
v=
1
4
3
Select the correct choice below and fill in the answer box to complete your choice.
(Type an integer or a simplified fraction.)
The vectors u and v are orthogonal because
uv=0.
The vectors u and v are not orthogonal because
uv=.
The vectors u and v are not orthogonal because
u+v=.
The vectors u and v are orthogonal because
u+v=.
6.1.16
Determine if the following vectors are orthogonal.
u=
10
5
15
,
v=
1
5
1
Select the correct choice below and fill in the answer box to complete your choice.
(Type an integer or a simplified fraction.)
The vectors u and v are orthogonal because
uv=0.
The vectors u and v are orthogonal because
u+v=.
The vectors u and v are not orthogonal because
u+v=.
The vectors u and v are not orthogonal because
uv=.
6.1.18
Determine if the following vectors are orthogonal.
u=
4
4
9
0
,
v=
1
8
16
4
Are the two vectors orthogonal?
(Type an integer or a fraction.)
The vectors u and v
are
orthogonal because
u+v=.
The vectors u and v
are
orthogonal because
uv=.
The vectors u and v
are not
orthogonal because
uv=108.
The vectors u and v
are not
orthogonal because
u+v=.
6.1.19
Determine whether the statement below is true or false. Justify the answer. The vector is in
n.
vv=v2
Choose the correct answer below.
The statement is false. By the definition of the length of a vector
v,
v=vv.
It follows that
v2=(vv)2.
The statement is true. By the definition of the length of a vector
v,
v=vv.
It follows that
v2=(vv)2.
The statement is false. The expression
vv
represents a vector, whereas
v2
is a scalar.
The statement is true. By the definition of the length of a vector
v,
v=vv.
6.1.19
Determine whether the statement below is true or false. Justify the answer. The vector is in
n.
vv=v2
Choose the correct answer below.
The statement is true. By the definition of the length of a vector
v,
v=vv.
The statement is false. The expression
vv
represents a vector, whereas
v2
is a scalar.
The statement is false. By the definition of the length of a vector
v,
v=vv.
It follows that
v2=(vv)2.
The statement is true. By the definition of the length of a vector
v,
v=vv.
It follows that
v2=(vv)2.
6.1.22
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
If
u2+v2=u+v2,
then
u
and
v
are orthogonal.
Choose the correct answer below.
The statement is false. Two vectors
u
and
v
are orthogonal if
uv=0.
If
u2+v2=u+v2,
then
uv=1.
The statement is false. If
u2+v2=u+v2,
then
u
and
v
are orthogonal complements.
The statement is true. By the Pythagorean Theorem, two vectors
u
and
v
are orthogonal if and only if
u+v2=u2+v2.
The statement is true. By the definition of the inner product, two vectors
u
and
v
are orthogonal if and only if
u+v2=u2+v2.
6.1.22
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
If
u2+v2=u+v2,
then
u
and
v
are orthogonal.
Choose the correct answer below.
The statement is true. By the Pythagorean Theorem, two vectors
u
and
v
are orthogonal if and only if
u+v2=u2+v2.
The statement is true. By the definition of the inner product, two vectors
u
and
v
are orthogonal if and only if
u+v2=u2+v2.
The statement is false. Two vectors
u
and
v
are orthogonal if
uv=0.
If
u2+v2=u+v2,
then
uv=1.
The statement is false. If
u2+v2=u+v2,
then
u
and
v
are orthogonal complements.
6.1.22
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
If
u2+v2=u+v2,
then
u
and
v
are orthogonal.
Choose the correct answer below.
The statement is true. By the definition of the inner product, two vectors
u
and
v
are orthogonal if and only if
u+v2=u2+v2.
The statement is true. By the Pythagorean Theorem, two vectors
u
and
v
are orthogonal if and only if
u+v2=u2+v2.
The statement is false. If
u2+v2=u+v2,
then
u
and
v
are orthogonal complements.
The statement is false. Two vectors
u
and
v
are orthogonal if
uv=0.
If
u2+v2=u+v2,
then
uv=1.
6.1.23
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
If vectors
v1,
...,
vp
span a subspace W and if
x
is orthogonal to each
vj
for
j=1,
..., p, then
x
is in
W.
Choose the correct answer below.
The statement is true. A vector
x
is in
W
if and only if
dist(x,v)=0
for any vector
v
in a set V that spans W.
The statement is true. If
x
is orthogonal to each
vj,
then
x
is also orthogonal to any linear combination of those
vj.
Since any vector in W can be described as a linear combination of
vj,
x
is orthogonal to all vectors in W.
The statement is false. It is known that a vector
x
is in
W
if and only if
x
is not orthogonal to every vector in a set that spans W. The vectors
v1,
...,
vp
make up a set V that spans the subspace W. This means
x
is orthogonal to every vector in V and, therefore,
x
is not in
W.
The statement is false. It is known that a vector
x
is in
W
if and only if
x
is orthogonal to every vector
w
in W. The vectors
v1,
...,
vp
make up a set V that spans the subspace W, but this is not enough to say that
x
is in
W.
6.1.23
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
If vectors
v1,
...,
vp
span a subspace W and if
x
is orthogonal to each
vj
for
j=1,
..., p, then
x
is in
W.
Choose the correct answer below.
The statement is true. A vector
x
is in
W
if and only if
dist(x,v)=0
for any vector
v
in a set V that spans W.
The statement is false. It is known that a vector
x
is in
W
if and only if
x
is not orthogonal to every vector in a set that spans W. The vectors
v1,
...,
vp
make up a set V that spans the subspace W. This means
x
is orthogonal to every vector in V and, therefore,
x
is not in
W.
The statement is false. It is known that a vector
x
is in
W
if and only if
x
is orthogonal to every vector
w
in W. The vectors
v1,
...,
vp
make up a set V that spans the subspace W, but this is not enough to say that
x
is in
W.
The statement is true. If
x
is orthogonal to each
vj,
then
x
is also orthogonal to any linear combination of those
vj.
Since any vector in W can be described as a linear combination of
vj,
x
is orthogonal to all vectors in W.
6.1.24
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
If
x
is orthogonal to every vector in a subspace W, then
x
is in
W.
Choose the correct answer below.
The statement is false. If
x
is orthogonal to every vector in a subspace W, then
x
is in W and so
x
cannot be in
W.
The statement is true. If
x
is orthogonal to every vector in a subspace W, then
x=0.
The zero vector is in every subspace, so
x
must be in
W.
The statement is false. A vector
x
is in
W
if and only if
x
is orthogonal to every vector in a set that spans W.
The statement is true. If
x
is orthogonal to every vector in W, then
x
is said to be orthogonal to W. The set of all vectors
x
that are orthogonal to W is denoted
W.
6.1.24
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
If
x
is orthogonal to every vector in a subspace W, then
x
is in
W.
Choose the correct answer below.
The statement is true. If
x
is orthogonal to every vector in W, then
x
is said to be orthogonal to W. The set of all vectors
x
that are orthogonal to W is denoted
W.
The statement is false. If
x
is orthogonal to every vector in a subspace W, then
x
is in W and so
x
cannot be in
W.
The statement is true. If
x
is orthogonal to every vector in a subspace W, then
x=0.
The zero vector is in every subspace, so
x
must be in
W.
The statement is false. A vector
x
is in
W
if and only if
x
is orthogonal to every vector in a set that spans W.
6.1.24
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
If
x
is orthogonal to every vector in a subspace W, then
x
is in
W.
Choose the correct answer below.
The statement is false. A vector
x
is in
W
if and only if
x
is orthogonal to every vector in a set that spans W.
The statement is true. If
x
is orthogonal to every vector in W, then
x
is said to be orthogonal to W. The set of all vectors
x
that are orthogonal to W is denoted
W.
The statement is true. If
x
is orthogonal to every vector in a subspace W, then
x=0.
The zero vector is in every subspace, so
x
must be in
W.
The statement is false. If
x
is orthogonal to every vector in a subspace W, then
x
is in W and so
x
cannot be in
W.
6.1.27
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
For a square matrix A, vectors in
Col A
are orthogonal to vectors in
Nul A.
Choose the correct answer below.
The statement is false. By the theorem of orthogonal complements,
(Row A)=Nul A.
This means that vectors in
Row A
are orthogonal to vectors in
Nul A,
so vectors in
Col A
cannot also be orthogonal to vectors in
Nul A.
The statement is false. By the theorem of orthogonal complements, it is known that vectors in
Col A
are orthogonal to vectors in
Nul AT.
Using the definition of orthogonal complements, vectors in
Col A
are orthogonal to vectors in
Nul A
if and only if the rows and columns of A are the same, which is not necessarily true.
The statement is true. By the theorem of orthogonal complements, it is known that vectors in
Col A
are orthogonal to vectors in
Nul AT.
Using the definition of orthogonal complements, vectors in
Col A
are orthogonal to vectors in
Nul A
if and only if the rows and columns of A are the same, which is true for any square matrix.
The statement is true. By the theorem of orthogonal complements,
(Row A)=Nul A.
By the definition of orthogonal complements, it follows that vectors in
Col A
are orthogonal to vectors in
Nul A.
6.1.27
Determine whether the statement below is true or false. Justify the answer. The vectors are in
n.
For a square matrix A, vectors in
Col A
are orthogonal to vectors in
Nul A.
Choose the correct answer below.
The statement is false. By the theorem of orthogonal complements, it is known that vectors in
Col A
are orthogonal to vectors in
Nul AT.
Using the definition of orthogonal complements, vectors in
Col A
are orthogonal to vectors in
Nul A
if and only if the rows and columns of A are the same, which is not necessarily true.
The statement is false. By the theorem of orthogonal complements,
(Row A)=Nul A.
This means that vectors in
Row A
are orthogonal to vectors in
Nul A,
so vectors in
Col A
cannot also be orthogonal to vectors in
Nul A.
The statement is true. By the theorem of orthogonal complements,
(Row A)=Nul A.
By the definition of orthogonal complements, it follows that vectors in
Col A
are orthogonal to vectors in
Nul A.
The statement is true. By the theorem of orthogonal complements, it is known that vectors in
Col A
are orthogonal to vectors in
Nul AT.
Using the definition of orthogonal complements, vectors in
Col A
are orthogonal to vectors in
Nul A
if and only if the rows and columns of A are the same, which is true for any square matrix.
6.2.2
Determine whether the set of vectors is orthogonal.
2
1
1
,
1
2
0
,
2
1
5
Is the set of vectors orthogonal?
The set of vectors is orthogonal because each pair of distinct vectors from the set is orthogonal.
The set of vectors is not orthogonal because at least one pair of distinct vectors from the set is not orthogonal.
The set of vectors is orthogonal because
2
1
1
is orthogonal to
1
2
0
and
1
2
0
is orthogonal to
2
1
5
.
The set of vectors is orthogonal because the vectors are linearly independent.
6.2.2
Determine whether the set of vectors is orthogonal.
4
1
1
,
1
4
0
,
4
1
17
Is the set of vectors orthogonal?
The set of vectors is orthogonal because
4
1
1
is orthogonal to
1
4
0
and
1
4
0
is orthogonal to
4
1
17
.
The set of vectors is orthogonal because the vectors are linearly independent.
The set of vectors is orthogonal because each pair of distinct vectors from the set is orthogonal.
The set of vectors is not orthogonal because at least one pair of distinct vectors from the set is not orthogonal.
6.2.2
Determine whether the set of vectors is orthogonal.
2
1
1
,
1
2
0
,
2
1
5
Is the set of vectors orthogonal?
The set of vectors is orthogonal because each pair of distinct vectors from the set is orthogonal.
The set of vectors is orthogonal because
2
1
1
is orthogonal to
1
2
0
and
1
2
0
is orthogonal to
2
1
5
.
The set of vectors is orthogonal because the vectors are linearly independent.
The set of vectors is not orthogonal because at least one pair of distinct vectors from the set is not orthogonal.
6.2.3
Determine whether the set of vectors is orthogonal.
2
5
1
,
8
4
4
,
3
1
1
Is the set of vectors orthogonal?
No
Yes
6.2.3
Determine whether the set of vectors is orthogonal.
1
3
1
,
0
1
3
,
10
3
1
Is the set of vectors orthogonal?
Yes
No
6.2.4
Determine whether the set of vectors is orthogonal.
4
5
7
,
0
0
0
,
2
4
4
Is the set of vectors orthogonal?
Yes
No
6.2.4
Determine whether the set of vectors is orthogonal.
0
0
0
,
2
3
7
,
4
2
2
Is the set of vectors orthogonal?
No
Yes
6.2.7
Show that
u1, u2
is an orthogonal basis for
2.
Then express x as a linear combination of the
u's.
u1=
2
4
,
u2=
8
4
,
and
x=
6
7
Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for
2?
Select all that apply.
The vectors must form an orthogonal set.
The vectors must all have a length of 1.
The vectors must span
2.
The distance between any pair of distinct vectors must be constant.
Which theorem could help prove one of these criteria from another?
If
S=u1, ..., up
is an orthogonal set of nonzero vectors in
n,
then S is linearly independent and hence is a basis for the subspace spanned by S.
If
S=u1, ..., up
is a basis in
p,
then the members of S span
p
and hence form an orthogonal set.
If
S=u1, ..., up
and each
ui
has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S.
If
S=u1, ..., up
and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.
What calculation shows that
u1, u2
is an orthogonal basis for
2?
Since the
inner product of u1 and u2
is
0,
the vectors
form an orthogonal set.
From the theorem above, this proves that the vectors are also
a basis for 2
because they are two
linearly independent vectors in 2.
Express x as a linear combination of the
u's.
x=2u1+14u2
(Simplify your answers. Use integers or fractions for any numbers in the equation.)
6.2.7
Show that
u1, u2
is an orthogonal basis for
2.
Then express x as a linear combination of the
u's.
u1=
2
6
,
u2=
12
4
,
and
x=
2
2
Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for
2?
Select all that apply.
The vectors must span
2.
The vectors must form an orthogonal set.
The distance between any pair of distinct vectors must be constant.
The vectors must all have a length of 1.
Which theorem could help prove one of these criteria from another?
If
S=u1, ..., up
and each
ui
has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S.
If
S=u1, ..., up
is a basis in
p,
then the members of S span
p
and hence form an orthogonal set.
If
S=u1, ..., up
and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.
If
S=u1, ..., up
is an orthogonal set of nonzero vectors in
n,
then S is linearly independent and hence is a basis for the subspace spanned by S.
What calculation shows that
u1, u2
is an orthogonal basis for
2?
Since the
inner product of u1 and u2
is
0,
the vectors
form an orthogonal set.
From the theorem above, this proves that the vectors are also
a basis for 2
because they are two
linearly independent vectors in 2.
Express x as a linear combination of the
u's.
x=25u1+110u2
(Simplify your answers. Use integers or fractions for any numbers in the equation.)
6.2.10
Show that
u1, u2, u3
is an orthogonal basis for
3.
Then express x as a linear combination of the
u's.
u1=
4
4
0
,
u2=
2
2
1
,
u3=
1
1
4
,
and
x=
4
3
1
Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for a subspace W of
n?
Select all that apply.
The distance between any pair of distinct vectors must be constant.
The vectors must all have a length of 1.
The vectors must form an orthogonal set.
The vectors must span W.
Which theorem could help prove one of these criteria from another?
If
S=u1, ..., up
and each
ui
has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S.
If
S=u1, ..., up
is a basis in
p,
then the members of S span
p
and hence form an orthogonal set.
If
S=u1, ..., up
is an orthogonal set of nonzero vectors in
n,
then S is linearly independent and hence is a basis for the subspace spanned by S.
If
S=u1, ..., up
and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.
Which calculations should be performed next?
(Simplify your answers.)
u1u2=
u1u3=
u2u3=
u1u2=0
u1u3=0
u2u3=0
u1u1=
u2u2=
u3u3=
How do these calculations show that
u1, u2, u3
is an orthogonal basis for
3?
Since each
inner product
is
0,
the vectors
form an orthogonal set.
From the theorem above, this proves that the vectors are also
a basis.
Express x as a linear combination of the
u's.
x=78u1+19u2+518u3
(Use integers or fractions for any numbers in the equation.)
6.2.10
Show that
u1, u2, u3
is an orthogonal basis for
3.
Then express x as a linear combination of the
u's.
u1=
3
3
0
,
u2=
2
2
1
,
u3=
1
1
4
,
and
x=
5
2
1
Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for a subspace W of
n?
Select all that apply.
The distance between any pair of distinct vectors must be constant.
The vectors must all have a length of 1.
The vectors must span W.
The vectors must form an orthogonal set.
Which theorem could help prove one of these criteria from another?
If
S=u1, ..., up
and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.
If
S=u1, ..., up
is an orthogonal set of nonzero vectors in
n,
then S is linearly independent and hence is a basis for the subspace spanned by S.
If
S=u1, ..., up
is a basis in
p,
then the members of S span
p
and hence form an orthogonal set.
If
S=u1, ..., up
and each
ui
has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S.
Which calculations should be performed next?
(Simplify your answers.)
u1u2=0
u1u3=0
u2u3=0
u1u1=
u2u2=
u3u3=
u1u2=
u1u3=
u2u3=
How do these calculations show that
u1, u2, u3
is an orthogonal basis for
3?
Since each
inner product
is
0,
the vectors
form an orthogonal set.
From the theorem above, this proves that the vectors are also
a basis.
Express x as a linear combination of the
u's.
x=76u1+59u2+718u3
(Use integers or fractions for any numbers in the equation.)
6.2.10
Show that
u1, u2, u3
is an orthogonal basis for
3.
Then express x as a linear combination of the
u's.
u1=
4
4
0
,
u2=
3
3
1
,
u3=
1
1
6
,
and
x=
4
3
1
Which of the following criteria are necessary for a set of vectors to be an orthogonal basis for a subspace W of
n?
Select all that apply.
The distance between any pair of distinct vectors must be constant.
The vectors must form an orthogonal set.
The vectors must all have a length of 1.
The vectors must span W.
Which theorem could help prove one of these criteria from another?
If
S=u1, ..., up
and the distance between any pair of distinct vectors is constant, then the vectors are evenly spaced and hence form an orthogonal set.
If
S=u1, ..., up
is an orthogonal set of nonzero vectors in
n,
then S is linearly independent and hence is a basis for the subspace spanned by S.
If
S=u1, ..., up
is a basis in
p,
then the members of S span
p
and hence form an orthogonal set.
If
S=u1, ..., up
and each
ui
has length 1, then S is an orthogonal set and hence is a basis for the subspace spanned by S.
Which calculations should be performed next?
(Simplify your answers.)
u1u1=
u2u2=
u3u3=
u1u2=0
u1u3=0
u2u3=0
u1u2=
u1u3=
u2u3=
How do these calculations show that
u1, u2, u3
is an orthogonal basis for
3?
Since each
inner product
is
0,
the vectors
form an orthogonal set.
From the theorem above, this proves that the vectors are also
a basis.
Express x as a linear combination of the
u's.
x=78u1+219u2+738u3
(Use integers or fractions for any numbers in the equation.)
6.2.11
Compute the orthogonal projection of
1
7
onto the line through
3
9
and the origin.
The orthogonal projection is
2
6
.
(Simplify your answer.)
6.2.11
Compute the orthogonal projection of
8
9
onto the line through
3
6
and the origin.
The orthogonal projection is
2
4
.
(Simplify your answer.)
6.2.13
Let
y=
1
6
and
u=
5
7
.
Write
y
as the sum of two orthogonal vectors, one in
Span {u}
and one orthogonal to
u.
y=y+z=
52
72
+
72
52
(Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)
6.2.13
Let
y=
1
4
and
u=
3
5
.
Write
y
as the sum of two orthogonal vectors, one in
Span {u}
and one orthogonal to
u.
y=y+z=
32
52
+
52
32
(Type an integer or simplified fraction for each matrix element. List the terms in the same order as they appear in the original list.)
6.2.15
Let
y=
4
7
and
u=
6
8
.
Compute the distance from y to the line through u and the origin.
The distance from y to the line through u and the origin is
1.
(Simplify your answer.)
6.2.15
Let
y=
9
6
and
u=
8
6
.
Compute the distance from y to the line through u and the origin.
The distance from y to the line through u and the origin is
35.
(Simplify your answer.)
6.2.15
Let
y=
4
2
and
u=
8
6
.
Compute the distance from y to the line through u and the origin.
The distance from y to the line through u and the origin is
45.
(Simplify your answer.)
6.2.17
Determine whether the set of vectors is orthonormal. If the set is only orthogonal, normalize the vectors to produce an orthonormal set.
u1=
23
13
13
 and u2=
13
0
23
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
The set of vectors is orthogonal only. The normalized vectors for
u1
and
u2
are
26
16
16
and
15
0
25
,
respectively.
(Type exact answers, using radicals as needed.)
The set of vectors is orthonormal.
The set of vectors is not orthogonal.
6.2.17
Determine whether the set of vectors is orthonormal. If the set is only orthogonal, normalize the vectors to produce an orthonormal set.
u1=
23
13
13
 and u2=
15
0
25
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
The set of vectors is orthogonal only. The normalized vectors for
u1
and
u2
are
23
16
16
and
55
0
255
,
respectively.
(Type exact answers, using radicals as needed.)
The set of vectors is orthonormal.
The set of vectors is not orthogonal.
6.2.19
Determine if the set of vectors is orthonormal. If the set is only orthogonal, normalize the vectors to produce an orthonormal set.
u=
0.6
0.8
,
v=
0.8
0.6
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
The set of vectors is orthogonal only. The normalized vectors for
u
and
v
are
and
,
respectively.
(Type an exact answer, using radicals as needed.)
The set of vectors is orthonormal.
The set of vectors is not orthogonal.
6.2.19
Determine if the set of vectors is orthonormal. If the set is only orthogonal, normalize the vectors to produce an orthonormal set.
u=
0.8
0.6
,
v=
0.6
0.8
Select the correct choice below and, if necessary, fill in the answer boxes to complete your choice.
The set of vectors is orthogonal only. The normalized vectors for
u
and
v
are
and
,
respectively.
(Type an exact answer, using radicals as needed.)
The set of vectors is orthonormal.
The set of vectors is not orthogonal.
6.2.23
Determine whether the statement below is true or false. Justify the answer.
Not every linearly independent set in
n
is an orthogonal set.
Choose the correct answer below.
The statement is true. For example, the vectors
0
1
and
1
1
are linearly independent but not orthogonal.
The statement is true. For example, the vectors
1
1
and
1
1
are linearly independent but not orthogonal.
The statement is false. Every orthogonal set is linearly independent.
The statement is false. In every linearly independent set of two vectors in
n,
one vector is a multiple of the other, so the vectors cannot be orthogonal.
6.2.23
Determine whether the statement below is true or false. Justify the answer.
Not every linearly independent set in
n
is an orthogonal set.
Choose the correct answer below.
The statement is true. For example, the vectors
0
1
and
1
1
are linearly independent but not orthogonal.
The statement is false. In every linearly independent set of two vectors in
n,
one vector is a multiple of the other, so the vectors cannot be orthogonal.
The statement is false. Every orthogonal set is linearly independent.
The statement is true. For example, the vectors
1
1
and
1
1
are linearly independent but not orthogonal.
6.2.23
Determine whether the statement below is true or false. Justify the answer.
Not every linearly independent set in
n
is an orthogonal set.
Choose the correct answer below.
The statement is false. In every linearly independent set of two vectors in
n,
one vector is a multiple of the other, so the vectors cannot be orthogonal.
The statement is true. For example, the vectors
0
1
and
1
1
are linearly independent but not orthogonal.
The statement is false. Every orthogonal set is linearly independent.
The statement is true. For example, the vectors
1
1
and
1
1
are linearly independent but not orthogonal.
6.2.24
Determine whether the statement below is true or false. Justify the answer.
Not every orthogonal set in
n
is linearly independent.
Choose the correct answer below.
The statement is true. Every orthogonal set of nonzero vectors is linearly independent, but not every orthogonal set is linearly independent.
The statement is true. Orthogonal sets with fewer than n vectors in
n
are not linearly independent.
The statement is false. Every orthogonal set of nonzero vectors is linearly independent and zero vectors cannot exist in orthogonal sets.
The statement is false. Orthogonal sets must be linearly independent in order to be orthogonal.
6.2.24
Determine whether the statement below is true or false. Justify the answer.
Not every orthogonal set in
n
is linearly independent.
Choose the correct answer below.
The statement is false. Orthogonal sets must be linearly independent in order to be orthogonal.
The statement is true. Orthogonal sets with fewer than n vectors in
n
are not linearly independent.
The statement is false. Every orthogonal set of nonzero vectors is linearly independent and zero vectors cannot exist in orthogonal sets.
The statement is true. Every orthogonal set of nonzero vectors is linearly independent, but not every orthogonal set is linearly independent.
6.2.25
Determine whether the statement below is true or false. Justify the answer. All vectors are in
n.
If
y
is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.
Choose the correct answer below.
The statement is false. The weights in any linear combination can only be computed using row operations.
The statement is true. If the orthogonal set is
u1, ..., up,
then for each
y,
the weights in the linear combination
y=c1u1++cpup
can be computed by
cj=yujujuj,
where
j=1, ..., p.
The statement is false. The weights in a linear combination can only be computed without row operations on a matrix if one of the vectors is the zero vector.
The statement is true. If the orthogonal set is
u1, ..., up,
then for each
y,
the weights in the linear combination
y=c1u1++cpup
can be computed by
cj=yujyy,
where
j=1, ..., p.
6.2.25
Determine whether the statement below is true or false. Justify the answer. All vectors are in
n.
If
y
is a linear combination of nonzero vectors from an orthogonal set, then the weights in the linear combination can be computed without row operations on a matrix.
Choose the correct answer below.
The statement is false. The weights in any linear combination can only be computed using row operations.
The statement is true. If the orthogonal set is
u1, ..., up,
then for each
y,
the weights in the linear combination
y=c1u1++cpup
can be computed by
cj=yujyy,
where
j=1, ..., p.
The statement is true. If the orthogonal set is
u1, ..., up,
then for each
y,
the weights in the linear combination
y=c1u1++cpup
can be computed by
cj=yujujuj,
where
j=1, ..., p.
The statement is false. The weights in a linear combination can only be computed without row operations on a matrix if one of the vectors is the zero vector.
6.2.26
Determine whether the statement below is true or false. Justify the answer. All vectors are in
n.
If a set
S=u1, ..., up
has the property that
uiuj=0
whenever
ij,
then S is an orthonormal set.
Choose the correct answer below.
The statement is true. To be orthonormal, the vectors in S must be orthogonal to each other.
The statement is false. To be orthonormal, S must be linearly independent of the subspace spanned by S.
The statement is true. Because
uiuj=0
whenever
ij,
S is linearly independent and thus is an orthonormal set.
The statement is false. To be orthonormal, the vectors in S must be unit vectors as well as being orthogonal to each other.
6.2.26
Determine whether the statement below is true or false. Justify the answer. All vectors are in
n.
If a set
S=u1, ..., up
has the property that
uiuj=0
whenever
ij,
then S is an orthonormal set.
Choose the correct answer below.
The statement is true. To be orthonormal, the vectors in S must be orthogonal to each other.
The statement is true. Because
uiuj=0
whenever
ij,
S is linearly independent and thus is an orthonormal set.
The statement is false. To be orthonormal, the vectors in S must be unit vectors as well as being orthogonal to each other.
The statement is false. To be orthonormal, S must be linearly independent of the subspace spanned by S.
6.2.28
Determine whether the statement below is true or false. Justify the answer. The vector is in
n.
If the columns of an
m×n
matrix A are orthonormal, then the linear mapping
xAx
preserves lengths.
Choose the correct answer below.
The statement is true. If A is a matrix with orthonormal columns, then
Ax=x.
The statement is false. Only nonzero orthonormal matrices preserve lengths.
The statement is true. If A is a matrix with orthonormal columns, then A must be a square matrix. This means the linear mapping
xAx
must preserve lengths.
The statement is false. If A is a matrix with orthonormal columns, then the linear mapping
xAx
would only preserve lengths if A is square.
6.2.28
Determine whether the statement below is true or false. Justify the answer. The vector is in
n.
If the columns of an
m×n
matrix A are orthonormal, then the linear mapping
xAx
preserves lengths.
Choose the correct answer below.
The statement is true. If A is a matrix with orthonormal columns, then
Ax=x.
The statement is false. If A is a matrix with orthonormal columns, then the linear mapping
xAx
would only preserve lengths if A is square.
The statement is false. Only nonzero orthonormal matrices preserve lengths.
The statement is true. If A is a matrix with orthonormal columns, then A must be a square matrix. This means the linear mapping
xAx
must preserve lengths.
6.2.29
Determine whether the statement below is true or false. Justify the answer.
A matrix with orthonormal columns is an orthogonal matrix.
Choose the correct answer below.
The statement is false. A matrix with orthonormal columns is an orthogonal matrix if the matrix is not square.
The statement is true. All matrices with orthonormal columns are orthogonal matrices.
The statement is true. All matrices with orthonormal rows and columns are orthogonal matrices.
The statement is false. A matrix with orthonormal columns is an orthogonal matrix if the matrix is also square.
6.2.29
Determine whether the statement below is true or false. Justify the answer.
A matrix with orthonormal columns is an orthogonal matrix.
Choose the correct answer below.
The statement is false. A matrix with orthonormal columns is an orthogonal matrix if the matrix is not square.
The statement is true. All matrices with orthonormal columns are orthogonal matrices.
The statement is false. A matrix with orthonormal columns is an orthogonal matrix if the matrix is also square.
The statement is true. All matrices with orthonormal rows and columns are orthogonal matrices.
6.2.32
Determine whether the statement below is true or false. Justify the answer.
An orthogonal matrix is invertible.
Choose the correct answer below.
The statement is true. An orthogonal matrix is a square invertible matrix U such that
U1=UT.
The statement is false. Only nonzero orthogonal matrices are invertible.
The statement is true. An orthogonal matrix is a square invertible matrix U such that
U1=U.
The statement is false. Only orthogonal matrices with orthonormal rows are invertible.
6.3.4
Verify that
u1,u2
is an orthogonal set, and then find the orthogonal projection of y onto
Spanu1,u2.
y=
4
5
4
,
u1=
4
5
0
,
u2=
5
4
0
To verify that
u1,u2
is an orthogonal set, find
u1u2.
u1u2=0
(Simplify your answer.)
The projection of y onto
Spanu1,u2
is
4
5
0
.
(Simplify your answers.)
6.3.4
Verify that
u1,u2
is an orthogonal set, and then find the orthogonal projection of y onto
Spanu1,u2.
y=
5
3
2
,
u1=
4
5
0
,
u2=
5
4
0
To verify that
u1,u2
is an orthogonal set, find
u1u2.
u1u2=0
(Simplify your answer.)
The projection of y onto
Spanu1,u2
is
5
3
0
.
(Simplify your answers.)
6.3.4
Verify that
u1,u2
is an orthogonal set, and then find the orthogonal projection of y onto
Spanu1,u2.
y=
3
5
1
,
u1=
3
6
0
,
u2=
6
3
0
To verify that
u1,u2
is an orthogonal set, find
u1u2.
u1u2=0
(Simplify your answer.)
The projection of y onto
Spanu1,u2
is
3
5
0
.
(Simplify your answers.)
6.3.8
Let W be the subspace spanned by
u1
and
u2,
and write
y
as the sum of a vector in W and a vector orthogonal to W.
y=
6
4
7
,
u1=
1
1
2
,
u2=
3
9
3
The sum is
y=y+z,
where
y=
1
5
3
is in W and
z=
7
1
4
is orthogonal to W.
(Simplify your answers.)
6.3.8
Let W be the subspace spanned by
u1
and
u2,
and write
y
as the sum of a vector in W and a vector orthogonal to W.
y=
2
3
7
,
u1=
1
1
3
,
u2=
2
5
1
The sum is
y=y+z,
where
y=
65
4
285
is in W and
z=
165
1
75
is orthogonal to W.
(Simplify your answers.)
6.3.8
Let W be the subspace spanned by
u1
and
u2,
and write
y
as the sum of a vector in W and a vector orthogonal to W.
y=
7
2
9
,
u1=
1
1
3
,
u2=
2
5
1
The sum is
y=y+z,
where
y=
1
92
112
is in W and
z=
8
52
72
is orthogonal to W.
(Simplify your answers.)
6.3.9
Let W be a subspace spanned by the
u's,
and write
y
as the sum of a vector in W and a vector orthogonal to W.
y=
4
2
2
1
,
u1=
1
1
0
1
,
u2=
1
5
3
4
,
u3=
1
0
1
1
y=
4017
5517
117
1017
+
2817
2117
3517
717
(Type an integer or simplified fraction for each matrix element.)
6.3.9
Let W be a subspace spanned by the
u's,
and write
y
as the sum of a vector in W and a vector orthogonal to W.
y=
4
6
6
1
,
u1=
1
1
0
1
,
u2=
1
5
3
4
,
u3=
1
0
1
1
y=
8851
13117
16151
2251
+
11651
2917
14551
2951
(Type an integer or simplified fraction for each matrix element.)
6.3.12
Find the closest point to
y
in the subspace W spanned by
v1
and
v2.
y=
3
1
1
19
,
v1=
1
3
1
1
,
v2=
1
1
0
4
The closest point to
y
in W is the vector
2
2
2
18
.
(Simplify your answer.)
6.3.12
Find the closest point to
y
in the subspace W spanned by
v1
and
v2.
y=
7
1
1
14
,
v1=
1
1
1
2
,
v2=
2
2
0
2
The closest point to
y
in W is the vector
3
3
5
12
.
(Simplify your answer.)
6.3.13
Find the best approximation to
z
by vectors of the form
c1v1+c2v2.
z=
3
6
2
3
,
v1=
2
1
2
1
,
v2=
1
1
0
1
The best approximation to
z
is
215
3910
15
3910
.
(Simplify your answer.)
6.3.13
Find the best approximation to
z
by vectors of the form
c1v1+c2v2.
z=
3
6
2
3
,
v1=
3
1
2
2
,
v2=
1
1
0
1
The best approximation to
z
is
136
6118
259
439
.
(Simplify your answer.)
6.3.13
Find the best approximation to
z
by vectors of the form
c1v1+c2v2.
z=
4
5
2
4
,
v1=
4
2
3
2
,
v2=
1
1
0
1
The best approximation to
z
is
5311
4511
411
4511
.
(Simplify your answer.)
6.3.15
Let
y=
6
8
5
,
u1=
2
5
1
,
u2=
3
1
1
.
Find the distance from y to the plane in
3
spanned by
u1
and
u2.
The distance is
73110.
(Type an exact answer, using radicals as needed.)
6.3.15
Let
y=
5
8
5
,
u1=
3
5
1
,
u2=
4
2
2
.
Find the distance from y to the plane in
3
spanned by
u1
and
u2.
The distance is
542.
(Type an exact answer, using radicals as needed.)
6.3.16
Let
y=
3
1
1
13
,
v1=
1
2
1
2
,
and
v2=
4
1
0
3
.
Find the distance from y to the subspace W of
4
spanned by
v1
and
v2,
given that the closest point to y in W is
y=
6
2
2
10
.
The distance is
6.
(Simplify your answer. Type an exact answer, using radicals as needed.)
6.3.16
Let
y=
3
1
1
13
,
v1=
1
2
1
2
,
and
v2=
4
1
0
3
.
Find the distance from y to the subspace W of
4
spanned by
v1
and
v2,
given that the closest point to y in W is
y=
6
2
2
10
.
The distance is
6.
(Simplify your answer. Type an exact answer, using radicals as needed.)
6.3.17
Let
y=
3
9
1
,
u1=
23
13
23
,
u2=
23
23
13
,
and
W=Span u1,u2.
Complete parts (a) and (b).
a. Let
U=
u1u2
.
Compute
UTU
and
UUT.
UTU=
10
01
and
UUT=
892929
295949
294959
(Simplify your answers.)
b. Compute
projWy
and
UUTy.
projWy=
409
559
359
and
UUTy=
409
559
359
(Simplify your answers.)
6.3.17
Let
y=
4
8
1
,
u1=
23
23
13
,
u2=
23
13
23
,
and
W=Span u1,u2.
Complete parts (a) and (b).
a. Let
U=
u1u2
.
Compute
UTU
and
UUT.
UTU=
10
01
and
UUT=
892929
295949
294959
(Simplify your answers.)
b. Compute
projWy
and
UUTy.
projWy=
2
4
5
and
UUTy=
2
4
5
(Simplify your answers.)
6.3.19
Let
u1=
2
2
2
,
u2=
2
1
1
,
and
u3=
0
0
1
.
Note that
u1
and
u2
are orthogonal but that
u3
is not orthogonal to
u1
or
u2.
It can be shown that
u3
is not in the subspace W spanned by
u1
and
u2.
Use this fact to construct a nonzero vector
v
in
3
that is orthogonal to
u1
and
u2.
A nonzero vector in
3
that is orthogonal to
u1
and
u2
is
v=
0
12
12
.
6.3.19
Let
u1=
1
1
1
,
u2=
2
1
1
,
and
u3=
0
0
1
.
Note that
u1
and
u2
are orthogonal but that
u3
is not orthogonal to
u1
or
u2.
It can be shown that
u3
is not in the subspace W spanned by
u1
and
u2.
Use this fact to construct a nonzero vector
v
in
3
that is orthogonal to
u1
and
u2.
A nonzero vector in
3
that is orthogonal to
u1
and
u2
is
v=
0
12
12
.
6.3.19
Let
u1=
2
1
1
,
u2=
3
3
3
,
and
u3=
0
0
1
.
Note that
u1
and
u2
are orthogonal but that
u3
is not orthogonal to
u1
or
u2.
It can be shown that
u3
is not in the subspace W spanned by
u1
and
u2.
Use this fact to construct a nonzero vector
v
in
3
that is orthogonal to
u1
and
u2.
A nonzero vector in
3
that is orthogonal to
u1
and
u2
is
v=
0
12
12
.
6.3.21
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
If
z
is orthogonal to
u1
and
u2
and if
W=Spanu1, u2,
then
z
must be in
W.
Choose the correct answer below.
The statement is false. Since
z
is orthogonal to
u1
and
u2,
it exists in
Spanu1, u2.
Since
W=Spanu1, u2,
z
is in W and cannot be in
W.
The statement is false. If
z
is orthogonal to
u1
and
u2,
it only follows that
z
is orthogonal to
Spanu1
and
Spanu2.
This is not enough information to conclude that
z
is in
W.
The statement is true. Since
z
is orthogonal to
u1
and
u2,
it is orthogonal to every vector in
Spanu1, u2,
a set that spans W.
The statement is true.
W
is the set of all vectors orthogonal to
u1
and
u2,
so by definition,
z
is in
W.
6.3.21
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
If
z
is orthogonal to
u1
and
u2
and if
W=Spanu1, u2,
then
z
must be in
W.
Choose the correct answer below.
The statement is false. Since
z
is orthogonal to
u1
and
u2,
it exists in
Spanu1, u2.
Since
W=Spanu1, u2,
z
is in W and cannot be in
W.
The statement is true. Since
z
is orthogonal to
u1
and
u2,
it is orthogonal to every vector in
Spanu1, u2,
a set that spans W.
The statement is false. If
z
is orthogonal to
u1
and
u2,
it only follows that
z
is orthogonal to
Spanu1
and
Spanu2.
This is not enough information to conclude that
z
is in
W.
The statement is true.
W
is the set of all vectors orthogonal to
u1
and
u2,
so by definition,
z
is in
W.
6.3.22
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
For each
y
and each subspace W, the vector
yprojWy
is orthogonal to W.
Choose the correct answer below.
The statement is false. Because
y
can be written uniquely in the form
y=projWy+z,
where
z
is in W and
projWy
is in
W,
it follows that
z=yprojWy.
The statement is true. Because
y
can be written uniquely in the form
y=projWy+z,
where
projWy
is in W and
z
is in
W,
it follows that
z=yprojWy.
The statement is true. Because
y
and
projWy
are both orthogonal to W, a linear combination of them must also be orthogonal to W.
The statement is false. Because
yprojWy
is in W, it cannot be orthogonal to W.
6.3.22
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
For each
y
and each subspace W, the vector
yprojWy
is orthogonal to W.
Choose the correct answer below.
The statement is true. Because
y
and
projWy
are both orthogonal to W, a linear combination of them must also be orthogonal to W.
The statement is true. Because
y
can be written uniquely in the form
y=projWy+z,
where
projWy
is in W and
z
is in
W,
it follows that
z=yprojWy.
The statement is false. Because
yprojWy
is in W, it cannot be orthogonal to W.
The statement is false. Because
y
can be written uniquely in the form
y=projWy+z,
where
z
is in W and
projWy
is in
W,
it follows that
z=yprojWy.
6.3.23
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
The orthogonal projection
y
of
y
onto a subspace W can sometimes depend on the orthogonal basis for W used to compute
y.
Choose the correct answer below.
The statement is false. The uniqueness property of the orthogonal decomposition
y=y+z
indicates that, no matter the basis used to find it, the decomposition will always be the same.
The statement is false. The orthogonal projection
y
of
y
onto a subspace W depends on an orthonormal basis for W.
The statement is true. For each possible orthogonal basis of W,
y
is expressed as a different linear combination of the vectors in that basis.
The statement is true. The orthogonal projection
y
of
y
onto a subspace W depends on an orthonormal basis for W.
6.3.24
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
If
y
is in a subspace W, then the orthogonal projection of
y
onto W is
y
itself.
Choose the correct answer below.
The statement is true. For an orthogonal basis
B=u1, ..., up
of W,
y
and
projWy
can be written as linear combinations of vectors in B with equal weights.
The statement is true. If
y
is in W, then
projWy=y,
which is in the same spanning set as
y.
The statement is false. If
y
is in W, then
projWy=0.
This means the statement is false unless
y=0.
The statement is false. If
y
is in W, then
projWy
is orthogonal to
y
and is in
W.
6.3.24
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
If
y
is in a subspace W, then the orthogonal projection of
y
onto W is
y
itself.
Choose the correct answer below.
The statement is false. If
y
is in W, then
projWy=0.
This means the statement is false unless
y=0.
The statement is true. For an orthogonal basis
B=u1, ..., up
of W,
y
and
projWy
can be written as linear combinations of vectors in B with equal weights.
The statement is false. If
y
is in W, then
projWy
is orthogonal to
y
and is in
W.
The statement is true. If
y
is in W, then
projWy=y,
which is in the same spanning set as
y.
6.3.25
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
The best approximation to
y
by elements of a subspace W is given by the vector
yprojWy.
The statement is
false.
The
Best Approximation Theorem
says that the best approximation to
y
is
projWy.
6.3.25
Determine whether the statement below is true or false. Justify the answer. Assume all vectors and subspaces are in
n.
The best approximation to
y
by elements of a subspace W is given by the vector
yprojWy.
The statement is
false.
The
Best Approximation Theorem
says that the best approximation to
y
is
projWy.
6.4.1
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.
0
8
8
, 
5
6
2
An orthogonal basis for W is
0
8
8
,
5
4
4
.
(Type a vector or list of vectors. Use a comma to separate vectors as needed.)
6.4.1
The given set is a basis for a subspace W. Use the Gram-Schmidt process to produce an orthogonal basis for W.
0
8
8
, 
8
8
4
An orthogonal basis for W is
0
8
8
,
8
6
6
.
(Type a vector or list of vectors. Use a comma to separate vectors as needed.)
6.4.10
Find an orthogonal basis for the column space of the matrix to the right.
   
156
362
216
152
   
An orthogonal basis for the column space of the given matrix is
1
3
2
1
,
3
0
3
3
,
2
0
0
2
.
(Type a vector or list of vectors. Use a comma to separate vectors as needed.)
6.4.10
Find an orthogonal basis for the column space of the matrix to the right.
   
165
284
127
143
   
An orthogonal basis for the column space of the given matrix is
1
2
1
1
,
2
0
2
0
,
0
2
0
4
.
(Type a vector or list of vectors. Use a comma to separate vectors as needed.)
6.4.10
Find an orthogonal basis for the column space of the matrix to the right.
   
166
184
126
142
   
An orthogonal basis for the column space of the given matrix is
1
1
1
1
,
1
3
3
1
,
6
5
4
3
.
(Type a vector or list of vectors. Use a comma to separate vectors as needed.)
6.4.14
The columns of Q were obtained by applying the Gram-Schmidt process to the columns of A. Find an upper triangular matrix R such that
A=QR.
A=
23
57
22
41
, Q=
27114
57214
270
47314
R=
77
014
6.4.14
The columns of Q were obtained by applying the Gram-Schmidt process to the columns of A. Find an upper triangular matrix R such that
A=QR.
A=
23
57
22
44
, Q=
27529
57229
270
470
R=
77
029
6.4.15
An orthogonal basis for the column space of matrix A is
v1, v2, v3.
Use this orthogonal basis to find a QR factorization of
matrix A.
   A=
124
121
042
164
167
,   v1=
1
1
0
1
1
,   v2=
2
2
4
2
2
,   v3=
2
3
1
1
2
Q=
12232219
12232319
0432119
12232119
12232219
,
R=
287
04232
0019
(Type exact answers, using radicals as needed.)
6.4.17
Determine whether the following statement is true or false, where all the vectors and subspaces are in
n.
Justify the answer.
If
v1, v2, v3
is an orthogonal basis for
W,
then multiplying
v3
by a scalar c gives a new orthogonal basis
v1, v2, cv3.
Choose the correct answer below.
The statement is false. If the scale factor is negative it will not give a new orthogonal basis.
The statement is true. For example, when the vectors in an orthogonal set are normalized, they are multiplied by scalars, and the set of new vectors is orthogonal.
The statement is true. Multiplication by a scalar, c, will change the length of
v3,
but it will not change the relative angles between the vectors and so they will still be orthogonal.
The statement is false. If the scale factor is zero it will not give a new orthogonal basis.
6.4.17
Determine whether the following statement is true or false, where all the vectors and subspaces are in
n.
Justify the answer.
If
v1, v2, v3
is an orthogonal basis for
W,
then multiplying
v3
by a scalar c gives a new orthogonal basis
v1, v2, cv3.
Choose the correct answer below.
The statement is false. If the scale factor is zero it will not give a new orthogonal basis.
The statement is false. If the scale factor is negative it will not give a new orthogonal basis.
The statement is true. For example, when the vectors in an orthogonal set are normalized, they are multiplied by scalars, and the set of new vectors is orthogonal.
The statement is true. Multiplication by a scalar, c, will change the length of
v3,
but it will not change the relative angles between the vectors and so they will still be orthogonal.
6.4.21
Determine whether the following statement is true or false, where all the vectors and subspaces are in
n.
Justify the answer.
If
A=QR,
where Q has orthonormal columns, then
R=QTA.
Choose the correct answer below.
The statement is false. If Q is square, then
QT=Q1
and
R=Q1A.
However, if Q is not square, the matrix
Q1
does not exist and this statement is false.
The statement is false. Since Q has orthonormal columns,
QTQ=0.
So
QTA=QT(QR)=0.
The statement is true. Since Q has orthonormal columns,
QTQ=1.
So
QTA=QT(QR)=1R=R.
The statement is true. Since Q has orthonormal columns,
QTQ=I.
So
QTA=QT(QR)=IR=R.
6.4.21
Determine whether the following statement is true or false, where all the vectors and subspaces are in
n.
Justify the answer.
If
A=QR,
where Q has orthonormal columns, then
R=QTA.
Choose the correct answer below.
The statement is false. If Q is square, then
QT=Q1
and
R=Q1A.
However, if Q is not square, the matrix
Q1
does not exist and this statement is false.
The statement is true. Since Q has orthonormal columns,
QTQ=1.
So
QTA=QT(QR)=1R=R.
The statement is true. Since Q has orthonormal columns,
QTQ=I.
So
QTA=QT(QR)=IR=R.
The statement is false. Since Q has orthonormal columns,
QTQ=0.
So
QTA=QT(QR)=0.
6.5.1
Find a least-squares solution of
Ax=b
by (a) constructing the normal equations for
x
and (b) solving for
x.
A=
12
23
11
,
b=
2
1
3
a. Construct the normal equations for
x.
69
914
x1
x2
=
3
4
(Simplify your answers.)
b. Solve for
x.
x=
2
1
(Simplify your answer.)
6.5.1
Find a least-squares solution of
Ax=b
by (a) constructing the normal equations for
x
and (b) solving for
x.
A=
11
12
11
,
b=
1
2
3
a. Construct the normal equations for
x.
34
46
x1
x2
=
2
0
(Simplify your answers.)
b. Solve for
x.
x=
6
4
(Simplify your answer.)
6.5.3
Find a least-squares solution of
Ax=b
by (a) constructing the normal equations for
x
and (b) solving for
x.
A=
14
14
02
47
,
b=
5
1
3
4
a. Construct the normal equations for
x
without solving.
1820
2085
x=
20
6
(Simplify your answers.)
b. Solve for
x.
x=
158113
146565
(Simplify your answer.)
6.5.4
Find a least-squares solution of
Ax=b
by (a) constructing the normal equations for
x
and (b) solving for
x.
A=
11
11
11
,
b=
4
6
0
a. Construct the normal equations for
x.
31
13
x1
x2
=
10
2
(Simplify your answers.)
b. Solve for
x.
x=
4
2
(Simplify your answer.)
6.5.5
Describe all least-squares solutions of the equation
Ax=b.
A=
110
101
101
110
,
b=
3
11
3
1
The least-squares solutions are all vectors of the form
x=
7
5
0
+x3
1
1
1
,
with
x3
free.
(Simplify your answers.)
6.5.5
Describe all least-squares solutions of the equation
Ax=b.
A=
101
101
110
110
,
b=
2
10
1
3
The least-squares solutions are all vectors of the form
x=
6
4
0
+x3
1
1
1
,
with
x3
free.
(Simplify your answers.)
6.5.7
Compute the least-squares error associated with the least-squares solution
x
of
Ax=b.
A=
12
12
04
59
,
b=
4
1
2
4
,
x=
1,5131,154
3491,154
The least-squares error is
155771154.
(Type an exact answer, using radicals as needed.)
6.5.8
For
A=
12
11
11
and
b=
3
8
0
,
a least-squares solution of
Ax=b
is
x=
5
2
.
Compute the least-squares error associated with this solution.
The least-squares error is
14.
(Simplify your answer. Type an exact answer, using radicals as needed.)
6.5.10
Find (a) the orthogonal projection of b onto Col A and (b) a least-squares solution of
Ax=b.
A=
14
18
14
,
b=
11
1
3
a. The orthogonal projection of b onto Col A is
b=
7
1
7
.
(Simplify your answers.)
b. A least-squares solution of
Ax=b
is
x=
5
12
.
(Simplify your answers.)
6.5.13
Let
A=
97
51
97
,
b=
10
7
6
,
u=
9
5
,
and
v=
9
3
.
Compute
Au
and
Av,
and compare them with
b.
Could
u
possibly be a least-squares solution of
Ax=b?
(Answer this without computing a least-squares solution.)
Au=
46
50
46
(Simplify your answer.)
Av=
60
48
60
(Simplify your answer.)
Compare
Au
and
Av
with
b.
Could
u
possibly be a least-squares solution of
Ax=b?
Au
is closer to
b
than
Av
is. Thus,
u
could
possibly be a least-squares solution of
Ax=b.
Av
is closer to
b
than
Au
is. Thus,
u
cannot
possibly be a least-squares solution of
Ax=b.
Au
and
Av
are equally close to
b.
Thus, both
u
and
v
can be a least-squares solution of
Ax=b.
Au
and
Av
are equally close to
b.
Thus, neither
u
nor
v
can be a least-squares solution of
Ax=b.
6.5.13
Let
A=
89
61
89
,
b=
11
5
9
,
u=
5
3
,
and
v=
5
6
.
Compute
Au
and
Av,
and compare them with
b.
Could
u
possibly be a least-squares solution of
Ax=b?
(Answer this without computing a least-squares solution.)
Au=
13
33
13
(Simplify your answer.)
Av=
14
36
14
(Simplify your answer.)
Compare
Au
and
Av
with
b.
Could
u
possibly be a least-squares solution of
Ax=b?
Au
is closer to
b
than
Av
is. Thus,
u
could
possibly be a least-squares solution of
Ax=b.
Av
is closer to
b
than
Au
is. Thus,
u
cannot
possibly be a least-squares solution of
Ax=b.
Au
and
Av
are equally close to
b.
Thus, both
u
and
v
can be a least-squares solution of
Ax=b.
Au
and
Av
are equally close to
b.
Thus, neither
u
nor
v
can be a least-squares solution of
Ax=b.
6.5.15
Use the factorization
A=QR
to find the least-squares solution of
Ax=b.
A=
23
24
11
=
2313
2323
1323
35
01
b=
7
3
1
x=
4
1
(Simplify your answer.)
6.5.17
Determine whether the statement below is true or false. Justify the answer. Assume A is an
m×n
matrix and
b
is in
m.
The general least-squares problem is to find an
x
that makes
Ax
as close as possible to
b.
Choose the correct answer below.
The statement is false. The general least-squares problem attempts to find an
x
such that
Ax=b.
The statement is false. The general least-squares problem attempts to find an
x
that maximizes
bAx.
The statement is true. The general least-squares problem attempts to find an
x
that minimizes
bAx.
The statement is true. The general least-squares problem attempts to find an
x
such that
Ax=b.
6.5.18
Determine whether the statement below is true or false. Justify the answer. Assume A is an
m×n
matrix and
b
is in
m.
If
b
is in the column space of A, then every solution of
Ax=b
is a least-squares solution.
The statement is
true.
A least-squares solution,
x,
is a vector that makes
Ax
the closest point in
Col A
to
b.
Since
b
is in
Col A,
all solutions of
Ax=b
are
least-squares solutions.
6.5.19
Determine whether the statement below is true or false. Justify the answer. Assume A is an
m×n
matrix and
b
is in
m.
A least-squares solution of
Ax=b
is a vector
x
that satisfies
Ax=b,
where
b
is the orthogonal projection of
b
onto
Col A.
Choose the correct answer below.
The statement is false. Since
b
is not the closest point in
Col A
to
b,
Ax=b
is not necessarily consistent.
The statement is false. The equations
Ax=b
and
Ax=b
are not equivalent.
The statement is true. The equations
Ax=b
and
Ax=b
are equivalent.
The statement is true. Since
b
is the closest point in
Col A
to
b,
Ax=b
is consistent and
x
is such that
Ax=b
is a least-squares solution of
Ax=b.
6.5.20
Determine whether the statement below is true or false. Justify the answer. Assume A is an
m×n
matrix and
b
is in
m.
A least-squares solution of
Ax=b
is a vector
x
such that
bAxbAx
for all
x
in
n.
Choose the correct answer below.
The statement is true because the general least-squares problem attempts to find an
x
that maximizes
bAx.
The statement is false because a least-squares solution of
Ax=b
is a vector
x
such that
bAx<bAx
for all
x
in
n.
The statement is false because a least-squares solution of
Ax=b
is a vector
x
such that
bAxbAx
for all
x
in
n.
The statement is true because the general least-squares problem attempts to find an
x
that minimizes
bAx.
6.5.21
Determine whether the statement below is true or false. Justify the answer. Assume A is an
m×n
matrix and
b
is in
m.
Any solution of
ATAx=ATb
is a least-squares solution of
Ax=b.
Choose the correct answer below.
The statement is true. The equation
ATAx=ATb
is equivalent to
Ax=b.
The statement is false. The set of least-squares solutions of
Ax=b
coincides with the nonempty set of solutions of the normal equations, defined as
ATx=b.
The statement is false. The set of least-squares solutions of
Ax=b
coincides with the nonempty set of solutions of the normal equations, defined as
ATx=ATAb.
The statement is true. The set of least-squares solutions of
Ax=b
coincides with the nonempty set of solutions of the normal equations, defined as
ATAx=ATb.
6.5.22
Determine whether the statement below is true or false. Justify the answer. Assume A is an
m×n
matrix and
b
is in
m.
If the columns of A are linearly independent, then the equation
Ax=b
has exactly one least-squares solution.
Choose the correct answer below.
The statement is true. If the columns of A are linearly independent, then
ATA
is invertible and
x=ATA1ATb
is the least-squares solution to
Ax=b.
The statement is false. If the columns of A are linearly independent, then there are infinitely many least-squares solutions to
Ax=b.
The statement is true. If the columns of A are linearly independent, then
AAT
is invertible and
x=AAT1ATb
is the least-squares solution to
Ax=b.
The statement is false. The least-squares solution is only guaranteed to be unique if the columns of A are orthogonal.
6.6.1
Find the equation
y=β0+β1x
of the least-squares line that best fits the given data points.
(1,2),
(2,2),
(3,4),
(4,4)
The line is
y=1+45x.
(Type integers or decimals.)
6.6.3
Find the equation
y=β0+β1x
of the least-squares line that best fits the given data points.
(1,0),
(0,1),
(1,2),
(2,4)
The line is
y=1110+1310x.
(Type integers or decimals.)
6.6.9
If you enter the following data into a machine and it returns a y value of
19
when
x=2.5,
should you trust the machine? Justify your answer.
(1,2),
(2,2),
(3,3),
(4,3)
Choose the correct answer below.
Yes, the machine should be trusted. Machine learning models are not limited to straight lines. A fifth or greater degree polynomial could intersect all four of the original data points and still pass through the predicted point at
(2.5,
19).
No, the machine should not be trusted. Since the data are not collinear, a linear model will not make any sense. A prediction like
y=19,
which is so far from the range of y values in the data is the kind of nonsensical prediction that results from using poor data.
Yes, the machine should be trusted. Although the predicted value of
y=19
may seem surprisingly large, machines very rarely make calculation errors.
No, the machine should not be trusted.
x=2.5
is within the range of x values in the data, while
y=19
is significantly greater than the maximum y value in the data. It is unreasonable to expect any model to predict a value so far outside the range of y values
6.6.13
A certain experiment produces the data
(1, 1.7),
(2, 2.8),
(3, 3.3),
(4, 3.6),
and
(5, 3.9).
Describe the model that produces a least-squares fit of these points by a function of the form
y=β1x+β2x2.
Such a function might arise, for example, as the revenue from the sale of x units of a product, when the amount offered for sale affects the price to be set for the product. Answer parts (a) through (c) below.
a. Give the design matrix, the observation vector, and the unknown parameter vector. Choose the correct design matrix X below.
11
24
39
416
525
11111
12345
1491625
111
124
139
1416
1525
12345
1491625
The observation vector is
y=
1.7
2.8
3.3
3.6
3.9
.
Choose the correct form of the parameter vector
β
below.
β1
β2
β3
β4
β5
0
β1
β2
x
x2
β1
β2
b. Find the associated least-squares curve for the data.
y=1.72x+0.19x2
(Round to two decimal places as needed.)
c. If a machine learned the curve you found in (b), what output would it provide for an input of
x=6?
y=3.48
(Round to two decimal places as needed.)
*9.1.B1
Determine the following sums or differences.
(a)
(9+5i)+(2+7i)
(b)
(5+8i)(74i)
(a)
(9+5i)+(2+7i)=11+12i
(Simplify your answer. Type your answer in the form
a+bi.)
(b)
(5+8i)(74i)=12+12i
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B1
Determine the following sums or differences.
(a)
(57i)+(39i)
(b)
(4+2i)(75i)
(a)
(57i)+(39i)=816i
(Simplify your answer. Type your answer in the form
a+bi.)
(b)
(4+2i)(75i)=11+7i
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B1
Determine the following sums or differences.
(a)
(17i)+(69i)
(b)
(58i)(7+6i)
(a)
(17i)+(69i)=716i
(Simplify your answer. Type your answer in the form
a+bi.)
(b)
(58i)(7+6i)=1214i
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B2
Express the following products in standard form.
(a)
(3+4i)(39i)
(b)
(5i)(5i)
(a)
(3+4i)(39i)=27+39i
(Simplify your answer. Type your answer in the form
a+bi.)
(b)
(5i)(5i)=26+0i
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B2
Express the following products in standard form.
(a)
(6+2i)(37i)
(b)
(3i)(3i)
(a)
(6+2i)(37i)=4+48i
(Simplify your answer. Type your answer in the form
a+bi.)
(b)
(3i)(3i)=10+0i
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B2
Express the following products in standard form.
(a)
(4+3i)(84i)
(b)
(3i)(3i)
(a)
(4+3i)(84i)=20+40i
(Simplify your answer. Type your answer in the form
a+bi.)
(b)
(3i)(3i)=10
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B3
Determine the complex conjugates of the following numbers.
(a)
17i
(b)
3
(c)
1112i
(d)
14+13i
(a) The complex conjugate of
17i
is
017i.
(Simplify your answer. Type your answer in the form
a+bi.)
(b) The complex conjugate of
3
is
3+0i.
(Simplify your answer. Type your answer in the form
a+bi.)
(c) The complex conjugate of
1112i
is
11+12i.
(Simplify your answer. Type your answer in the form
a+bi.)
(d) The complex conjugate of
14+13i
is
1413i.
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B3
Determine the complex conjugates of the following numbers.
(a)
18i
(b)
10
(c)
64i
(d)
3+5i
(a) The complex conjugate of
18i
is
018i.
(Simplify your answer. Type your answer in the form
a+bi.)
(b) The complex conjugate of
10
is
10+0i.
(Simplify your answer. Type your answer in the form
a+bi.)
(c) The complex conjugate of
64i
is
6+4i.
(Simplify your answer. Type your answer in the form
a+bi.)
(d) The complex conjugate of
3+5i
is
35i.
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B3
Determine the complex conjugates of the following numbers.
(a)
19i
(b)
4
(c)
1611i
(d)
15+5i
(a) The complex conjugate of
19i
is
19i.
(Simplify your answer. Type your answer in the form
a+bi.)
(b) The complex conjugate of
4
is
4.
(Simplify your answer. Type your answer in the form
a+bi.)
(c) The complex conjugate of
1611i
is
16+11i.
(Simplify your answer. Type your answer in the form
a+bi.)
(d) The complex conjugate of
15+5i
is
155i.
(Simplify your answer. Type your answer in the form
a+bi.)
*9.1.B5
Express the following quotients in standard form.
(a)
348i
(b)
16i4+8i
(a)
348i=320+310i
(Simplify your answer. Type your answer in the form
a+bi.
Use integers or fractions for any numbers in the expression.)
(b)
16i4+8i=112025i
(Simplify your answer. Type your answer in the form
a+bi.
Use integers or fractions for any numbers in the expression.)
*9.1.B5
Express the following quotients in standard form.
(a)
678i
(b)
59i7+8i
(a)
678i=42113+48i113
(Simplify your answer. Type your answer in the form
a+bi.
Use integers or fractions for any numbers in the expression.)
(b)
59i7+8i=37113103113i
(Simplify your answer. Type your answer in the form
a+bi.
Use integers or fractions for any numbers in the expression.)
*9.1.B5
Express the following quotients in standard form.
(a)
958i
(b)
46i5+8i
(a)
958i=4589+7289i
(Simplify your answer. Type your answer in the form
a+bi.
Use integers or fractions for any numbers in the expression.)
(b)
46i5+8i=28896289i
(Simplify your answer. Type your answer in the form
a+bi.
Use integers or fractions for any numbers in the expression.)
*9.1.B6
Use polar form to determine
z1z2
and
z1z2.
z1=13i,
z2=3i
Use polar form to determine
z1z2.
Choose the correct answer below.
z1z2=18cos14π+isin14π
z1z2=4cos56π+isin56π
z1z2=4cos12π+isin12π
z1z2=cos56π+isin56π
Use polar form to determine
z1z2.
Choose the correct answer below.
z1z2=322cos14π+isin14π
z1z2=cos12π+isin12π
z1z2=cos56π+isin56π
z1z2=4cos12π+isin12π
*9.1.B6
Use polar form to determine
z1z2
and
z1z2.
z1=1i,
z2=33i
Use polar form to determine
z1z2.
Choose the correct answer below.
z1z2=6cos12π+isin12π
z1z2=62cos14π+isin14π
z1z2=13(cos(π)+isin(π))
z1z2=6(cos(π)+isin(π))
Use polar form to determine
z1z2.
Choose the correct answer below.
z1z2=13(cos(π)+isin(π))
z1z2=cos14π+isin14π
z1z2=13cos12π+isin12π
z1z2=6cos12π+isin12π
*9.1.B6
Use polar form to determine
z1z2
and
z1z2.
z1=13i,
z2=33i
Use polar form to determine
z1z2.
Choose the correct answer below.
z1z2=62cos112π+isin112π
z1z2=62cos1712π+isin1712π
z1z2=4cos13π+isin13π
z1z2=232cos1712π+isin1712π
Use polar form to determine
z1z2.
Choose the correct answer below.
z1z2=232cos112π+isin112π
z1z2=62cos112π+isin112π
z1z2=232cos1712π+isin1712π
z1z2=cos13π+isin13π
*9.1.B7
Use polar form to determine the following.
(33i)3
(33i)3=5454i
(Simplify your answer. Type your answer in the form
a+bi.
Type an exact answer, using radicals as needed.)
*9.1.B7
Use polar form to determine the following.
3+i4
3+i4=8+83i
(Simplify your answer. Type your answer in the form
a+bi.
Type an exact answer, using radicals as needed.)
*9.1.B7
Use polar form to determine the following.
(33i)5
(33i)5=972+972i
(Simplify your answer. Type your answer in the form
a+bi.
Type an exact answer, using radicals as needed.)
*9.2.B1
For parts (a) and (b) below, determine whether the system is consistent, and if it is, determine the general solution.
(a)
z2iz3
=
5
iz1z2+(1i)z3
=
79i
2iz1+z2+(3+2i)z3
=
15+16i
Choose the correct answer below.
z=
1+2i
3+4i
4+2i
z=
1+2i
0
0
+s
3+4i
4+2i
1
z=
1+2i
3+4i
0
+s
4+2i
0
1
The system is inconsistent.
(b)
z1+(1i)z2+iz3
=
2i
iz1+(1i)z2+3iz4
=
i
z1+(1i)z2+(1+i)z3+3iz4
=
3+2i
Choose the correct answer below.
z=
1
0
1+3i
0
z=
1
0
1+3i
0
+s
1+i
1
0
0
+t
3
0
3i
1
z=
1
0
1+3i
0
+s
1+i
1
0
0
The system is inconsistent.
*9.3.B1
Calculate the following.
(a)
3+5i
5i
22i
43i
(b)
3i
55i
5+4i
(c)
(32i)
3+4i
14i
4+i
(a)
3+5i
5i
22i
43i
=
1+7i
4+8i
(Simplify your answer.)
(b)
3i
55i
5+4i
=
15+15i
12+15i
(Simplify your answer.)
(c)
(32i)
3+4i
14i
4+i
=
118i
5+14i
14+5i
(Simplify your answer.)
*9.3.B1
Calculate the following.
(a)
3i
24i
+
1+i
42i
(b)
3i
25i
1+2i
(c)
(2+4i)
4i
33i
3+2i
(a)
3i
24i
+
1+i
42i
=
2
66i
(Simplify your answer.)
(b)
3i
25i
1+2i
=
15+6i
6+3i
(Simplify your answer.)
(c)
(2+4i)
4i
33i
3+2i
=
12+14i
618i
148i
(Simplify your answer.)
*9.3.B1
Calculate the following.
(a)
3i
1+5i
4+5i
3+4i
(b)
4i
3+i
2+4i
(c)
(1+4i)
5i
25i
13i
(a)
3i
1+5i
4+5i
3+4i
=
48i
2+i
(Simplify your answer.)
(b)
4i
3+i
2+4i
=
412i
168i
(Simplify your answer.)
(c)
(1+4i)
5i
25i
13i
=
20+5i
18+13i
13i
(Simplify your answer.)
*9.3.B3
Determine which of the following sets is a basis for
3.
(a)
0
2i
0
, 
5
0
2
, 
0
2
i
(b)
3i
3i
3i
, 
0
0
i
, 
3
i
1
(a) Is the given set a basis for
3?
No, because it is a linearly independent set that spans
3.
Yes, because it is a linearly dependent set that spans
3.
No, because one of the vectors in the set is not in
3.
Yes, because it contains three vectors in
3.
Yes, because it is a linearly independent set that spans
3.
No, because it is a linearly dependent set that does not span
3.
(b) Is the given set a basis for
3?
Yes, because it is a linearly dependent set that spans
3.
Yes, because it contains three vectors in
3.
Yes, because it is a linearly independent set that spans
3.
No, because it is a linearly independent set that spans
3.
No, because only one of the vectors in the set is in
3.
No, because it is a linearly dependent set that does not span
3.
*9.3.B3
Determine which of the following sets is a basis for
3.
(a)
5
1
2
, 
0
2
2i
, 
1
2i
0
(b)
i
i
i
, 
1
1
i
, 
2
2i
1
(a) Is the given set a basis for
3?
No, because it is a linearly dependent set that does not span
3.
Yes, because it contains three vectors in
3.
No, because one of the vectors in the set is not in
3.
No, because it is a linearly independent set that spans
3.
Yes, because it is a linearly dependent set that spans
3.
Yes, because it is a linearly independent set that spans
3.
(b) Is the given set a basis for
3?
No, because it is a linearly dependent set that does not span
3.
No, because it is a linearly independent set that spans
3.
Yes, because it is a linearly dependent set that spans
3.
Yes, because it contains three vectors in
3.
Yes, because it is a linearly independent set that spans
3.
No, because only one of the vectors in the set is in
3.
*9.3.B3
Determine which of the following sets is a basis for
3.
(a)
3
2
0
, 
2
1
2i
, 
0
i
1
(b)
0
i
1
, 
0
2
i
, 
2i
2i
2i
(a) Is the given set a basis for
3?
No, because it is a linearly independent set that spans
3.
No, because one of the vectors in the set is not in
3.
Yes, because it contains three vectors in
3.
No, because it is a linearly dependent set that does not span
3.
Yes, because it is a linearly dependent set that spans
3.
Yes, because it is a linearly independent set that spans
3.
(b) Is the given set a basis for
3?
No, because it is a linearly independent set that spans
3.
No, because only one of the vectors in the set is in
3.
No, because it is a linearly dependent set that does not span
3.
Yes, because it contains three vectors in
3.
Yes, because it is a linearly independent set that spans
3.
Yes, because it is a linearly dependent set that spans
3.
*9.3.B4
Find a basis for the rowspace, columnspace, and nullspace of the following matrices.
(a)
A=
222i
2i2i2
1+i1+i1i
(b)
B=
0i22i
1+7i2i5
303i
(a) Determine a basis for Row(A). Choose the correct answer below.
1
0
0
,
0
1
0
,
0
0
1
1
1
0
,
0
0
1
1
0
0
,
0
1
1
1
2
i
Determine a basis for Col(A). Choose the correct answer below.
2i
2
1i
2
2i
1+i
,
2
2i
1+i
,
2i
2
1i
2
2i
1+i
,
2i
2
1i
2
2i
1+i
Determine a basis for Null(A). Choose the correct answer below.
0
1
1
1
1
0
2
1
0
,
i
0
1
The basis is the empty set.
(b) Determine a basis for Row(B). Choose the correct answer below.
1
0
0
,
0
1
0
,
0
0
1
1
11
11i
1
1
0
,
0
0
1
1
0
0
,
0
1
5
Determine a basis for Col(B). Choose the correct answer below.
0
1+7i
3
0
1+7i
3
,
i
2i
0
,
22i
5
3i
0
1+7i
3
,
22i
5
3i
0
1+7i
3
,
i
2i
0
Determine a basis for Null(B). Choose the correct answer below.
11
1
0
,
11i
0
1
1
1
0
0
1
5
The basis is the empty set.
*9.3.B4
Find a basis for the rowspace, columnspace, and nullspace of the following matrices.
(a)
A=
444i
4i4i4
2+2i2+2i22i
                                            (b)
B=
02i43i
2+6ii3
203i
(a) Determine a basis for Row(A). Choose the correct answer below.
1
4
2i
1
0
0
,
0
1
2
1
1
0
,
0
0
1
1
0
0
,
0
1
0
,
0
0
1
Determine a basis for Col(A). Choose the correct answer below.
4
4i
2+2i
4
4i
2+2i
,
4i
4
22i
4
4i
2+2i
,
4
4i
2+2i
,
4i
4
22i
4i
4
22i
Determine a basis for Null(A). Choose the correct answer below.
1
1
0
4
1
0
,
2i
0
1
0
2
1
The basis is the empty set.
(b) Determine a basis for Row(B). Choose the correct answer below.
1
2
0
,
0
0
1
1
11
9i
1
0
0
,
0
1
0
,
0
0
1
1
0
0
,
0
1
3
Determine a basis for Col(B). Choose the correct answer below.
0
2+6i
2
0
2+6i
2
,
2i
i
0
,
43i
3
3i
0
2+6i
2
,
2i
i
0
0
2+6i
2
,
43i
3
3i
Determine a basis for Null(B). Choose the correct answer below.
1
2
0
0
2
3
11
1
0
,
9i
0
1
The basis is the empty set.
*9.3.B4
Find a basis for the rowspace, columnspace, and nullspace of the following matrices.
(a)
A=
222i
2i2i2
1+i1+i1i
                 (b)
B=
02i33i
3+3i3i3
203i
(a) Determine a basis for Row(A). Choose the correct answer below.
1
0
0
,
0
1
1
1
2
i
1
0
0
,
0
1
0
,
0
0
1
1
1
0
,
0
0
1
Determine a basis for Col(A). Choose the correct answer below.
2
2i
1+i
,
2i
2
1i
2i
2
1i
2
2i
1+i
2
2i
1+i
,
2
2i
1+i
,
2i
2
1i
Determine a basis for Null(A). Choose the correct answer below.
1
1
0
2
1
0
,
i
0
1
0
1
1
The basis is the empty set.
(b) Determine a basis for Row(B). Choose the correct answer below.
1
11
8i
1
0
0
,
0
1
0
,
0
0
1
1
2
0
,
0
0
1
1
0
0
,
0
1
3
Determine a basis for Col(B). Choose the correct answer below.
0
3+3i
2
,
2i
3i
0
0
3+3i
2
,
2i
3i
0
,
33i
3
3i
0
3+3i
2
,
33i
3
3i
0
3+3i
2
Determine a basis for Null(B). Choose the correct answer below.
0
1
3
11
1
0
,
8i
0
1
1
2
0
The basis is the empty set.